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Necessary and sufficient conditions for admissible set open topologies
Topology and its Applications 256 (2019) 198–207
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Topology and its Applications www.elsevier.com/locate/topol
Necessary and sufficient conditions for admissible set open topologies Abderrahmane Bouchair ∗ , Imane Dekkar LMAM Laboratory, Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria
a r t i c l e
i n f o
Article history: Received 25 March 2018 Received in revised form 1 February 2019 Accepted 9 February 2019 Available online 12 February 2019 MSC: 54C35 Keywords: Function space Set open topology Admissible topology M -Space M -Flow
a b s t r a c t Let X and Y be topological spaces and Cα (X, Y ) be the set of all continuous functions from X to Y endowed with the set open topology where α is a nonempty family of subsets of X. We obtain criteria in order that set open topology on C(X, Y ) to be admissible. We prove that the topology of Cα (X, Y ) is admissible if and only if α is a regular family, when Y is an equiconnected topological space. In the case when X is a topological monoid, we study the relationship between the admissibility of the set open topology on C(X, Y ) and the continuity of an action of X on C(X, Y ). We prove that the set open topology on C(X, Y ) is admissible if and only if Cα (X, Y ) with the action s.f = (t → f (st)) is the cofree X-space over Y . © 2019 Elsevier B.V. All rights reserved.
1. Introduction Given two topological spaces X and Y we denote by C(X, Y ) the set of all continuous functions from X to Y . Let α be a family of subsets of X and Cα (X, Y ) be the set C(X, Y ) endowed with the set open topology. If τ is a topology on C(X, Y ) then the corresponding space is denoted by Cτ (X, Y ). A topology τ on the set C(X, Y ) is called admissible or jointly continuous if the evaluation mapping e : Cτ (X, Y ) × X → Y defined by e(f, x) = f (x) is continuous or equivalently, for every space Z, the continuity of a map f : Z −→ Cτ (X, Y ) implies that of the map fˆ : Z ×X −→ Y defined by fˆ(z, x) = f (z)(x) for every (z, x) ∈ Z × X. This notion was first discussed by Arens in [2] who proved that the compact open topology on C(X, Y ) is admissible when X is a regular locally compact space, and beyond that, this one is the least admissible topology on C(X, Y ). Afterwards, during their studies of splitting and admissible topologies on the spaces of continuous functions, Arens and Dugundji [3] showed that the uniform convergence topology
* Corresponding author. E-mail addresses: [email protected] (Abderrahmane Bouchair), [email protected] (Imane Dekkar). https://doi.org/10.1016/j.topol.2019.02.008 0166-8641/© 2019 Elsevier B.V. All rights reserved.
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is always admissible, they introduced also a class of topologies on C(X, Y ) by the help of a family α of subsets of X, called the set open topology, and they gave a sufficient condition for these topologies to be admissible. Since that time, many works have been focussed on the problem of the existence of admissible topologies on C(X, Y ). However, few of them provide necessary and sufficient conditions for the admissibility of these topologies, see for instance [10–12,17,18]. In the same context, Nokhrin [18] managed to prove that if the family α consists of closed subsets of X, then the set open topology on C(X, R) is admissible if and only if the family α is a regular family. Further properties concerning set open topologies are included in the papers [5], [13], [18] and [19]. Recently, Khosravi in [14] studied, for a topological monoid M , the cofree and injective M -spaces and M -flows over a completely regular (Tychonoff) space in the category, M -Tych, of all completely regular M -spaces with continuous M -morphisms. He established a relationship between the existence of the cofree object in the category M -Tych and the fact that the compact open topology on C(M, Y ) being admissible. It is worth mentioning that cofree objects, like all the other universal objects, are very important in categories because of their special property and thereby the question of the existence of such kind of objects in a category has been the main topic of several researches during the last decades. We refer the interested reader in this direction to [16]. Relying on the aforementioned works, our aim in the present paper is twofold. Firstly, we will give a necessary and sufficient condition for the admissibility of the set open topology on continuous function spaces in the case when the range space is an equiconnected space. Secondly, we will establish the interrelation between the admissibility of this last and the continuity of the actions of topological monoids on function spaces and obtain, by this, other necessary and sufficient conditions for the admissibility of this topology on C(M, Y ), in the case when M is a topological monoid. This paper is organized as follows. Section 2 contains notations and definitions that will be used in the paper. In Section 3, we prove that if α is a functional refinement family consisting of Y -compact subsets of a space X and Y is an equiconnected topological space with a base B consisting of Ψ-convex sets, then the set open topology on C(X, Y ) is admissible if and only if α is a regular family. In Section 4, we prove that if α is a compact network on a topological monoid M and Y is a topological space, then the set open topology on C(M, Y ) is admissible if and only if Cα (M, Y ) with the action s.f := (t → f (st)) is the cofree M -space over Y . In the last section, we study the existence of the cofree M -flow over Y , where M is a completely regular topological monoid and Y is a compact space. 2. Preliminaries Throughout this paper, X and Y are topological spaces and C(X, Y ) is the set of all continuous functions from X to Y . Let α be a nonempty family of subsets of X. The set open topology τα on C(X, Y ) has a subbase consisting of sets of the form [A, V ] = {f ∈ C (X) : f (A) ⊆ V }, where A ∈ α and V is an open subset of Y , and the set C(X, Y ) equipped with the set open topology is denoted by Cα (X, Y ). If V is not arbitrary but is restricted to some collection B of open subsets of Y , then we denote by CαB (X, Y ) the corresponding function space. Note that if α is replaced by the family of all finite unions of its elements, then the topology of Cα (X, Y ) does not change; so we can assume that the family α is closed with respect to finite unions. The symbols ∅ and N will stand for the empty set and the positive integers, respectively. We denote by R the real numbers with the usual topology. The interior and the closure of a subset A in X is denoted by intA and A, respectively. βX denotes the Čech–Stone compactification of the space X. A family α of subsets of X is called a network on X if for any point x ∈ X and any neighborhood U of x there is an A ∈ α such that x ∈ A ⊂ U . A compact network on X is a network in which each member of it is compact.
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It is proved in [17] that if α is a compact network on a topological space and Y is a completely regular space, then Cα (X, Y ) is a completely regular space. A family α of subsets of X is called a regular family [3] if for any point x ∈ X and for any neighborhood V of x, there exists an A ∈ α contained in V and containing x in its interior. Let X, Y and Z be topological spaces. The exponential function Λ : Y Z×X −→ (Y X )Z is defined by {[Λ(f )](z)}(x) = f (z, x) for each f : Z × X −→ Y , for each z ∈ Z and each x ∈ X. A topology τ on the set C(X, Y ) is called splitting or proper if for any space Z, the continuity of a map f : Z × X → Y implies that of the map Λ(f ) : Z → Cτ (X, Y ). Arens and Dugundji [3] proved that the compact open topology on C(X, Y ) is always splitting and if, in addition, X is a regular locally compact, then it coincides with the greatest splitting topology which always exists. It is also proved that if α is a family consisting of compact subsets of X, then the set open topology τα on C(X, Y ) is splitting. A topological space Y is said to be equiconnected [9] if there exists a continuous map Ψ : Y ×Y ×[0, 1] → Y such that Ψ(x, y, 0) = x, Ψ(x, y, 1) = y, and Ψ(x, x, t) = x for all x, y ∈ Y and t ∈ [0, 1]. The map Ψ is called an equiconnecting function. A subset V of an equiconnected space Y is called a Ψ-convex subset of Y if Ψ(V, V, [0, 1]) ⊆ V . It is a known fact that any topological vector space or any convex subset of any topological vector space is an equiconnected space and any equiconnected space is a pathwise connected space. Let M be a monoid and A be an arbitrary set. Recall that a set A is a (right) M -set if there is a right action λ : A × M −→ A such that a · s = λ(a, s), (a · s) · t = a · (st) and a · 1 = a. Let A and B be two M -sets. An M -homomorphism from A to B is a map f : A → B such that f (a · s) = f (a) · s, for each a ∈ A and for each s ∈ M . Let M be a monoid and A be an M -set. For any s ∈ M we define the M -homomorphism μs : A → A by μs (y) = y · s. We define also, for every a ∈ A, the M -homomorphism ρa : M → A by ρa (t) = a · t. In the (M ) (M ) case when A = M , we use the notation μs : M −→ M and ρs : M −→ M . Let M be a monoid endowed with a topology τM . M is called semitopological monoid if the multiplication (M ) (M ) mapping M × M −→ M is separately continuous. That is, the M -homomorphisms μs and ρs are continuous for every s ∈ M . If the multiplication mapping M × M −→ M is continuous, then M is called topological monoid. Let M be a topological monoid. An M -space is an M -set A with a topology τA such that the action A × M −→ A is continuous. Any compact M -space is called an M -flow. Let C be a concrete category over D and | , | : C −→ D be the forgetful functor. Let D be an object in D, A an object in C and ψ : |A| −→ D a morphism (in D). We say that A is the cofree object over D if for every object B in C and for every morphism h : |B| −→ D (in D), there exists a unique morphism h : B −→ A h| = h. Recall that in M -Act, the category of all M -sets and M -homomorphisms, in C such that ψ ◦ | the cofree M -set over a set Y is the set Y M of all mappings from M to Y with the action λ defined by λ(f, s) = f · s := (s −→ (f · s)(s ) = f (ss )), for every f ∈ Y M , and every s, s ∈ M . It is proved in [15] that in the category M -semiTop, where the objects are all semi-topological M -sets, the cofree object is Cp (M, Y ), the space C(M, Y ) equipped with the pointwise convergence topology. For standard terminology we refer to [8]. 3. Admissibility of set open topology on C(X,Y) for equiconnected space Y Throughout this section, X is a completely regular space. We will study the continuity and separate continuity of the evaluation mapping e : CαB (X, Y ) × X → Y when the range space Y is an equiconnected topological space. Recall that if x ∈ X, then the evaluation mapping at x is the map ex : C(X, Y ) → Y defined by ex (f ) = e(f, x) = f (x) for every f ∈ C(X, Y ). So, the separate continuity of the evaluation mapping e is equivalent to the continuity of all maps ex . It is clear that ex : Cp (X, Y ) → Y is continuous
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and so it is for any topology stronger than the topology of pointwise convergence on C(X, Y ). Therefore by Theorem 1.1.3 of [17], we have the following result. Corollary 3.1. Let X and Y be two topological spaces and α a family of subsets of X. If α is a network on X, then the evaluation mapping e : Cα (X, Y ) × X → Y is separately continuous. For the converse of the above corollary, we have the following theorem. Theorem 3.1. Let X be a topological space, α a family of closed subsets of X, and let Y be an equiconnected space with a base B consisting of Ψ-convex sets. The evaluation map e : CαB (X, Y ) × X → Y is separately continuous if and only if α is a network. Proof. Let x0 ∈ X and U an open neighborhood of it such that no F ∈ α containing x0 is contained in U . Let f : X → [0, 1] be a continuous function with f (x0 ) = 1 and f (X \ U ) = {0}. Take a nontrivial path p in Y with p(0) = p(1), an open basic set W ∈ B such that p(1) ∈ W ⊆ Y \ {p(0)}, and put g = p ◦ f . Then n g ∈ e−1 [Ai , Vi ], where Ai ∈ α and x0 (W ). Since ex0 is continuous, there exists an open neighborhood Og = i=1
Vi ∈ B for all i, of g such that Og ⊆ e−1 / Ai }. We have then I = ∅. If we suppose that x0 (W ). Put I = {i : x0 ∈ x0 belongs to Ai for each i ∈ {1, . . . , n}, then the continuous function h constantly equal to p(0) belongs to Og because p(0) ∈ Vi for each i from the fact that Ai U . But h does not belong to e−1 x0 (W ), which is a contradiction. So I = ∅. Then the set ∪i∈I Ai is closed and does not contain x0 . Since X is a completely regular space, there exists a continuous function ϕ : X → [0, 1] such that ϕ(x0 ) = 0 and ϕ( Ai ) = {1}. i∈I
Consider the function h1 : X → Y defined by h1 (t) = Ψ(p(0), g(t), ϕ(t)), for every t ∈ X. So h1 is continuous and belongs to Og . To see this last fact, take an arbitrary i ∈ {1, . . . , n} and take any point t ∈ Ai . If i ∈ I, then h1 (t) = g(t) ∈ Vi . If i ∈ / I, we get Ai ∩ (X \ U ) = ∅. Hence p(0) ∈ Vi . Ψ-convexity of Vi leads us to i∈I /
the fact that h1 (t) ∈ Vi . Whence h1 ∈ Og . On the other hand, we have h1 (x0 ) = p(0) ∈ / W . So h1 ∈ / e−1 x0 (W ) which contradicts the fact that ex0 is continuous. Hence, α is a network. 2 Recall that a family α of subsets of X is called a functional refinement [13] if for every A ∈ α, every finite n
sequence U1 , . . . , Un of open subsets of Y and every f ∈ C(X, Y ) such that A ⊆ ∪ f −1 (Ui ), there exists a i=1
finite sequence A1 , . . . , Am of members of α that refines f −1 (U1 ), . . . , f −1 (Un ) whose union contains A. A subset B of X is said to be Y -compact if for any continuous function f ∈ C(X, Y ), the set f (B) is compact in Y . Below, we give some well known properties of Y -compact sets. (1) For a Hausdorff space Y , any compact set is Y -compact in any space X. (2) There are Y -compact sets of a compact space that are not closed, see [19, Example 3.9]. (3) There are closed sets that are not Y -compact, see [19, Example 3.12]. It is proved [13] that if α is a functional refinement family consisting of Y -compact subsets of X and B is any arbitrary base for Y , then the family {[A, V ] : A ∈ α, V ∈ B} forms a subbase for the set open topology on C(X, Y ). That is, Cα (X, Y ) = CαB (X, Y ). So we have the following. Corollary 3.2. Let X be a topological space, α a functional refinement family consisting of closed Y -compact subsets of X and let Y be an equiconnected space with a base B consisting of Ψ-convex sets. Then the evaluation map e : Cα (X, Y ) × X → Y is separately continuous if and only if α is a network.
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To see under which conditions the set open topology on C(X, Y ) is admissible, we state the following result obtained by Arens and Dugundji in [3]. Theorem 3.2. [3, Theorem 4.7] Let X and Y be two topological spaces and α a family of subsets of X. If α is a regular family, then the set open topology τα on C(X, Y ) is an admissible topology. The converse of Theorem 3.2 is not true in general. Indeed, it is proved in [18, Example 3.8], that if X = [0, 1] and α is the family consisting of the point x = 0, all intervals [a, b] where 0 < a < b ≤ 1, and all sets of the form [0, a] ∪ {1}, then α is not a regular family. However the evaluation mapping e : Cα (X) × R → R is continuous. Note that, without any loss of generality, we assume that α contains finite unions of its members. Theorem 3.3. Let α be a family of closed subsets of X, and Y an equiconnected T1 -space with a base B consisting of Ψ-convex sets. Then the evaluation map e : CαB (X, Y ) × X → Y is continuous if and only if α is a regular family. Proof. We prove only the necessary condition. We will show that α is a regular family. To do this, let x0 be a point in X and U an open neighborhood of it. Since X is a completely regular space, there exists a continuous function f : X → [0, 1] such that f (x0 ) = 1 and f (X \ U ) = {0}. Take a path p in Y with p(0) = p(1), an open basic set W ∈ B such that p(1) ∈ W and p(0) ∈ / W , and put g = p ◦ f . Since the mapping e is continuous, there exists an open neighborhood V of x0 in X and an open neighborhood Og of g in CαB (X, Y ) such that e(Og × V ) ⊆ W . We have then V ⊆ U . Suppose, on the contrary, that there is x ∈ V \ U . Then e(g, x ) = g(x ) = p(0) ∈ / W , a contradiction. Therefore V ⊆ U . Let us take n Og = [Ai , Vi ], where Ai ∈ α and Vi ∈ B and put A = ∪{Ai : 1 ≤ i ≤ n and Ai ⊆ U }. We have then i=1
A = ∅. Indeed, suppose that Ai does not contain in U for each i. It follows that p(0) ∈ Vi , for each 1 ≤ i ≤ n. On the other hand, the continuous function h : X → Y defined by h(z) = p(0), for every z ∈ X, belongs to Og . But e(h, V ) = h(V ) ⊆ {p(0)} ⊂ W , which is a contradiction. Hence A = ∅. Now, let us prove that x0 ∈ intA. Suppose, on the contrary, that x0 ∈ / intA. As above, it is easy to check that there is at least i0 for which x0 ∈ / Ai0 . Let us put V = V ∩ (X \ ∪{Ai : x0 ∈ / Ai }). Clearly, V is an open neighborhood of x0 which is not contained in A. Take a point x1 ∈ V \ A and put I = {i : x1 ∈ / Ai }. Since X is a completely regular space, there is a continuous function ϕ : X → [0, 1] such that ϕ(x1 ) = 0 and ϕ( Ai ) = {1}. Then i∈I
the function h1 : X → Y defined by h1 (t) = Ψ(p(0), g(t), ϕ(t)), for every t ∈ X, is continuous and belongs to Og . Indeed, let i ∈ {1, . . . , n} and take any point t ∈ Ai . If i ∈ I, then h1 (t) = g(t) ∈ Vi . If i ∈ / I, since x1 ∈ Ai we have x0 ∈ Ai and Ai does not contain in A. So Ai ∩(X \U ) = ∅. Hence p(0) ∈ Vi . Ψ-convexity i∈I /
of Vi leads us to the fact that h1 (t) ∈ Vi . Whence h1 ∈ Og . On the other hand, we have e(h1 , x1 ) = p(0). This contradiction proves that x0 ∈ intA. Thus, the family α is a regular family. 2 Corollary 3.3. Let α be a functional refinement family consisting of closed Y -compact subsets of a topological space X and Y be an equiconnected T1 -space with a base B consisting of Ψ-convex sets. Then, the set open topology τα on C(X, Y ) is admissible if and only if α is a regular family. 4. Admissibility of set open topologies and cofree M -spaces over a topological space Let M be a topological monoid and Y a topological space. In this section, we present the relationship between the fact that set open topology on the function space C(M, Y ) being admissible and the continuity of the action of M on C(M, Y ).
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Theorem 4.1. Let M be a topological monoid, Y a topological space, and α be a network consisting of compact subsets of M . The following statements are equivalent. 1. The set open topology τα on C(M, Y ) is admissible. 2. Cα (M, Y ) with the action λ defined by λ(f, s) = f · s := (s −→ (f · s)(s ) = f (ss )), for every f ∈ Cα (M, Y ) and s, s ∈ M , is the cofree M -space over Y . Proof. (1) ⇒ (2) We will first prove that Cα (M, Y ) with the action λ mentioned above is an M -space. Afterwards, we shall show that is cofree over the space Y . We have that λ is an action. In fact, let f ∈ Cα (M, Y ) and s1 , s2 ∈ M . We have that for every s ∈ M [(f · s1 ) · s2 ](s ) = (f · s1 )(s2 s ) = f (s1 s2 s ) = [f · (s1 s2 )](s ), so (f · s1 ) · s2 = f · (s1 s2 ). Furthermore, we have (f · 1M )(s ) = f (1M s ) = f (s ), for every s ∈ M . Then f · 1M = f . Now let us prove that λ is continuous. Let f ∈ Cα (M, Y ), s ∈ M , B ∈ α and U be an open subset in Y such that f · s ∈ [B, U ]. We have f · s ∈ [B, U ] ⇔ ∀b ∈ B, (f · s)(b) ∈ U ⇔ ∀b ∈ B, e(f, sb) ∈ U. Since the set open topology on C(M, Y ) is admissible, the evaluation map e is continuous. Then for every b ∈ B there exist an open subset Wsb in M which contains sb and an open neighborhood Of of f in Cα (M, Y ) such that e(Of × Wsb ) ⊆ U.
(1)
Furthermore, since M is a topological monoid, there exist two open neighborhoods Wsb and Wb of s and b respectively, such that Wsb Wb ⊆ Wsb . n Since B ⊆ b∈B Wb and B is compact, there exists {b1 , b2 , . . . , bn } ⊂ B such that B ⊆ i=1 Wbi . Put n Ws = i=1 Wsbi , which is open. We have then Ws B ⊆
n
Wsbi
i=1
n
Wbi ⊆
i=1
n
Wsbi Wbi ⊆
i=1
n
Wsbi .
i=1
From (1) we get e(Of ×
n i=1
Wsbi ) =
n
e(Of × Wsbi ) ⊆ U.
i=1
Then e(Of × Ws B) ⊆ e(Of ×
n
Wsbi ) ⊆ U.
(2)
i=1
We have that Of × Ws is open in Cα (M, Y ) × M which contains (f, s). We will prove in the sequel that λ(Of × Ws ) = Of · Ws ⊆ [B, U ]. Let g ∈ Of · Ws . Then there exist f0 ∈ Of and s0 ∈ Ws such that g = f0 · s0 . Thus, for every b ∈ B, we obtain
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g(b) = (f0 · s0 )(b) = f0 (s0 b) = e(f0 , s0 b) ∈ e(Of × Ws B). From (2) we have g(b) ∈ U , and so g(B) ⊂ U . Therefore g ∈ [B, U ]. Thus, the action λ is continuous and hence Cα (M, Y ) is an M -space. It remains to show that is cofree over Y . Consider the map ψ : Cα (M, Y ) −→ Y defined by ψ(f ) = f (1M ), for every f ∈ C(M, Y ). This map is continuous. In fact, let f ∈ Cα (M, Y ) and U an open neighborhood of ψ(f ). We have 1M ∈ f −1 (U ). Since α is a network, there exists B0 ∈ α such that 1M ∈ B0 ⊂ f −1 (U ). We set W = [B0 , U ]. Then f ∈ W and ψ(W ) ⊂ U . This proves that ψ is continuous. Take an M -space A and a continuous map h : A −→ Y , and let us show that there exists a unique continuous M -homomorphism h = h. Consider the map h : A −→ Cα (M, Y ) such that ψ ◦ h from A to Cα (M, Y ) defined by h(a))(s ) = h(a · s )) h(a) := (s −→ ( for every a ∈ A and s ∈ M . Thus, for every s, s ∈ M , we have [ h(a) · s](s ) = [ h(a)](ss ) = h(a · (ss )) = h((a · s) · s )) = [ h(a · s)](s ). So h is an M -homomorphism. If we denote by λ1 the action of M over A, then the map h ◦ λ1 :
A×M (a, s)
−→ A −→ Y −→ a · s −→ h(a · s)
is continuous. Since the set open topology in this case is splitting because the family α consists of compact sets, then Λ(h ◦ λ1 ) is continuous. Furthermore, we have that for every a ∈ A and s ∈ M {[Λ(h ◦ λ1 )](a)}(s) = (h ◦ λ1 )(a, s) = h(λ1 (a, s)) = h(a · s) = [ h(a)](s). Hence Λ(h ◦ λ1 ) = h. This allows us to deduce the continuity of h. Besides, (ψ ◦ h)(a) = ψ( h(a)) = ( h(a))(1M ) = h(a · 1M ) = h(a), for every a ∈ A. Then ψ ◦ h = h. Now, suppose that there exists g : A −→ Cα (M, Y ) an M -homomorphism such that ψ ◦ g = h. We have then that for every a ∈ A and s ∈ M (g(a))(s) = (g(a))(s · 1M ) = (g(a) · s)(1M ) = ψ(g(a) · s) = (ψ ◦ g)(a · s) = h(a · s) = ( h(a))(s). Hence g = h, and so h is unique. (2) ⇒ (1) Since Cα (M, Y ) is an M -space and the map ψ : Cα (M, Y ) −→ Y defined above is continuous, the map ψ ◦ λ is continuous. Furthermore, for every (f, s) in Cα (M, Y ) × M we have (ψ ◦ λ)(f, s) = ψ(f · s) = (f · s)(1M ) = f (s) = e(f, s). That is, e = ψ ◦ λ. Thus, the evaluation mapping e is continuous which means that the set open topology on C(M, Y ) is admissible. 2
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Recall that a topological space is called an Alexandroff space [1] if the intersection of any family of open subsets of it is open. We have the following theorem. Theorem 4.2. Let M be an Alexandroff topological monoid, Y a topological space and α be a network on M . Then the following statements are equivalent. 1. The set open topology τα on C(M, Y ) is admissible. 2. Cα (M, Y ) with the action λ defined by λ(f, s) = f · s := (s −→ (f · s)(s ) = f (ss )), for every f ∈ Cα (M, Y ) and s, s ∈ M , is an M -space. Proof. (1) ⇒ (2) As in the proof of Theorem 4.1, λ is an action. For the continuity of λ we follow the same argument of this proof by taking Ws = b∈B Wsb , which is an open subset of M because it is an Alexandroff space. Hence Cα (M, Y ) is an M -space. (2) ⇒ (1) The map ψ : Cα (M, Y ) −→ Y defined by ψ(f ) = f (1M ), for every f ∈ C(M, Y ), is continuous with the same reasoning as in the proof of Theorem 4.1. Since Cα (M, Y ) with the action λ is an M -space, the map e = ψ ◦ λ is continuous. Hence the set open topology τα on C(M, Y ) is admissible. 2 Remark 4.1. It is useful to mention that we can not confirm here that Cα(M, Y ) is cofree over Y as in Theorem 4.1, this is because the set open topology in this case is not splitting. 5. Admissibility of set open topologies and cofree M -flows over a topological space Using Theorem 4.1 above we study, in this section, the existence of the cofree M -flow over Y , where M is a completely regular topological monoid and Y is a compact space. We denote by βCα (M, Y ) the Čech–Stone compactification of the function space Cα (M, Y ) and by β(Cα (M, Y ) × M ) the Čech–Stone compactification of the product space Cα (M, Y ) × M . Theorem 5.1. Let M be a completely regular topological monoid, Y a compact space and α be a compact network on M . The following statements are equivalent. 1. The set open topology τα on C(M, Y ) is admissible. 2. There exists an action : βCα (M, Y ) × M −→ βCα (M, Y ), λ |C (M,Y )×M coincides with the action of Cα (M, Y ) and βCα (M, Y ) is the cofree M -flow such that λ α over Y . Proof. (1) ⇒ (2) Suppose that the set open topology on C(M, Y ) is admissible. Then, by Theorem 4.1, then we shall prove that Cα (M, Y ) with the action λ is the cofree M -space over Y . Firstly we will define λ, it is continuous and finally we will show that βCα (M, Y ) is cofree over Y . Since α is a compact network and Y is compact, then Cα (M, Y ) is completely regular. This leads to the fact that β(Cα (M, Y ) × M ) exists and for the action λ : Cα (M, Y ) × M −→ Cα (M, Y ) there exists, by Corollary 3.6.6 in [8], a continuous ¯ : β(Cα (M, Y ) × M ) −→ βCα (M, Y ) such that λ ¯ |C (M,Y )×M = λ. map λ α Fix an arbitrary element t ∈ M and define the mapping k : Cα (M, Y ) −→ Cα (M, Y ) × M , such as k(f ) = (f, t), ∀f ∈ Cα (M, Y ).
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It is clear that this map is continuous and we have k(Cα (M, Y )) = {(f, t) : f ∈ Cα (M, Y )} = Cα (M, Y ) × {t}. By definition of compactification, k(Cα (M, Y )) ⊂ Cα (M, Y ) × M can be considered as a subspace of β(Cα (M, Y ) × M ) and, so, the closure of k(Cα (M, Y )) will be compact in β(Cα (M, Y ) × M ). Furthermore, there exists a compact space B such that the closure of k(Cα (M, Y )) in β(Cα (M, Y ) × M ) is equal to B × {t}. Let us denote by ϕ0 the projection ϕ0 : B × M −→ B. Using Corollary 3.6.6 from [8] once again, the mapping ϕ0 ◦k can be extended to a continuous mapping h : βCα (M, Y ) −→ B that makes the following diagram commutative β
−→ βCα (M, Y ) ↓h −→ B
Cα (M, Y ) k ↓ B×M
ϕ0
Put λ = λ|B×M and define := λ ◦ (h × idM ) : βCα (M, Y ) × M −→ B × M −→ βCα (M, Y ). λ is continuous and we have also that for each s ∈ M and every f ∈ Cα (M, Y ), Clearly λ ¯ |B×M (h(f ), s) = λ ¯ |B×M [(ϕ0 (f, t), s)] = λ ¯ |B×M (f, s) = λ(f, s). s) = [λ ¯ |B×M ◦ (h × idM )](f, s) = λ λ(f, |C (M,Y )×M = λ. Now, let us prove that λ is an action. Let g ∈ βCα (M, Y ) and s, s ∈ This means that λ α M . The space Cα (M, Y ) can be considered as a dense subspace in βCα (M, Y ). Then there exists a net (fi )i∈I ⊂ Cα (M, Y ) converging to g. We have λ(lim λ(g, s), s ). ss ) = lim λ(f i , ss ) = lim λ(fi , ss ) = lim λ( λ(f i , s), s ) = λ( fi , s), s ) = λ( λ(g, i
i
i
i
is an action. We deduce that βCα (M, Y ) is an M -space. Since it is compact, it is an M -flow. It Hence λ remains to show that βCα (M, Y ) is cofree over Y . The continuous function ψ : Cα (M, Y ) → Y defined by ψ(f ) = f (1M ), for every f ∈ Cα (M, Y ), can be extended to a continuous function ψ¯ : βCα (M, Y ) −→ Y . Let us take an M -flow F and a continuous function h : F −→ Y . Since Cα (M, Y ) is cofree over Y , there exists an unique continuous M -homomorphism h = h. Since h : F −→ Cα (M, Y ) such that ψ ◦ ψ¯|Cα (M,Y ) = ψ, we have that for every y ∈ F ¯ (ψ¯ ◦ h)(y) = ψ[ h(y)] = ψ[ h(y)] = (ψ ◦ h)(y) = h(y). on βCα (M, Y ) where βCα (M, Y ) with this action (2) ⇒ (1) Suppose that there exists a continuous action λ is an M -flow and λ|Cα (M,Y )×M = λ, with λ is the action of M on Cα (M, Y ). We have then Cα (M, Y ) is an M -space, and since ψ : Cα (M, Y ) −→ Y is continuous, then the mapping λ
ψ
ψ ◦ λ : Cα (M, Y ) × M −→ Cα (M, Y ) −→ Y is also continuous. Furthermore, we have for every f ∈ Cα (M, Y ) and s ∈ M (ψ ◦ λ)(f, s) = ψ(λ(f, s)) = ψ(f · s) = (f · s)(1M ) = f (s) = e(f, s). This means that e = ψ ◦ λ which implies that the evaluation mapping e is continuous. Hence the set open topology on Cα (M, Y ) is admissible. 2
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In the end, it is interesting to discuss the injectivity of cofree objects, in case of existence, in the category M -Tych and the one of M -flows. It is known that injectivity of objects is a very useful notion which category theory inherited from homological and commutative algebra [20], and it has been studied, significantly, in different categories such as the category Pos of partial ordered sets and monotone mappings, the category Top of topological spaces, the category Boo of boolean algebras and boolean homomorphisms . . . etc., see for example [4], [6], [7] and the references cited therein. Let us take a completely regular topological monoid M and a completely regular space Y . If α is a regular family of compact subsets of M , then the set open topology on C(M, Y ) is admissible and splitting [3]. Moreover, M is a regular locally compact space in this case and thus the compact open topology is admissible too and it coincides with the set open topology, i.e. Cα (M, Y ) = Ck (M, Y ), where Ck (M, Y ) denotes the set of continuous functions endowed with the compact open topology. Consequently, Ck (M, Y ) is the cofree M -space over Y . Here we note that Ck (M, Y ) is completely regular, and it is injective in the category M -Tych if and only if Y is injective in the category Tych (Proposition 3.3 of [14]). As well, Ck (M, Y ) is injective in the category Tych if and only if Ck (M, Y ) is injective in the category M-Tych (Proposition 3.5 of [14]). If in addition Y is a compact space, then the cofree M -flow over Y , βCk (M, Y ), is injective in the category of M -flows if and only if Y is injective in the category of compact spaces (Proposition 3.4 of [14]). Acknowledgements The authors would like to thank the anonymous referee for his/her careful reading and insightful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
F.G. Arenas, Alexandroff spaces, Acta Math. Univ. Comen. 1 (1999) 17–25. R. Arens, A topology for spaces of transformations, Ann. Math. 47 (1946) 480–495. R. Arens, J. Dugundji, Topologies for function spaces, Pac. J. Math. 1 (1951) 5–31. H. Barzegar, M.M. Ebrahimi, M. Mahmoudi, Essentiality and injectivity, Appl. Categ. Struct. 18 (2010) 73–83. A. Bouchair, S. Kelaiaia, Comparison of some set open topologies on C(X, Y ), Topol. Appl. 178 (2014) 352–359. F. Cagliari, S. Mantovani, Injective topological fibre spaces, Topol. Appl. 125 (2002) 525–532. F. Cagliari, S. Mantovani, Injectivity and sections, J. Pure Appl. Algebra 204 (2006) 79–89. R. Engelking, General Topology, Polish Scientific Publishing, 1977. R. Fox, On fibre spaces II, Bull. Am. Math. Soc. 49 (1943) 733–735. D. Georgiou, A. Megaritis, K. Papadopoulos, Admissible topologies on C(Y,Z) and OZ (Y ), Quest. Answ. Gen. Topol. 32 (2014) 17–33. D.N. Georgiou, S.D. Iliadis, On the greatest splitting topology, Topol. Appl. 156 (2008) 70–75. D.N. Georgiou, S.D. Iliadis, B.K. Papadopoulos, On dual topologies, Topol. Appl. 140 (2004) 57–68. L. Harkat, A. Bouchair, S. Kelaiaia, Comparison of some set open and uniform topologies and some properties of the restriction map, Hacet. J. Math. Stat. (2019), https://doi.org/10.15672/HJMS.2017.508. B. Khosravi, On cofree S-spaces and cofree S-flows, Appl. Gen. Topol. 13 (2012) 1–10. B. Khosravi, The category of semitopological S-acts, in: Special Issue of Appl. Math., World Appl. Sci. J. 7 (2009) 07-13. M. Kilp, U. Kanauer, A.V. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin, New York, 2000. R.A. McCoy, I. Ntantu, Topological Properties of Spaces of Continuous Functions, Lect. Notes Math., vol. 1315, SpringerVerlag, Berlin, Heidelberg, 1988. S.E. Nokhrin, Some properties of set-open topologies, J. Math. Sci. 144 (2007) 4123–4151. A.V. Osipov, The set open topology, Topol. Proc. 37 (2011) 205–217. T. Walter, Injectivity versus exponentiability, Cah. Topol. Géom. Différ. Catég. 49 (3) (2008) 228–240.
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Topology and its Applications www.elsevier.com/locate/topol
Necessary and sufficient conditions for admissible set open topologies Abderrahmane Bouchair ∗ , Imane Dekkar LMAM Laboratory, Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria
a r t i c l e
i n f o
Article history: Received 25 March 2018 Received in revised form 1 February 2019 Accepted 9 February 2019 Available online 12 February 2019 MSC: 54C35 Keywords: Function space Set open topology Admissible topology M -Space M -Flow
a b s t r a c t Let X and Y be topological spaces and Cα (X, Y ) be the set of all continuous functions from X to Y endowed with the set open topology where α is a nonempty family of subsets of X. We obtain criteria in order that set open topology on C(X, Y ) to be admissible. We prove that the topology of Cα (X, Y ) is admissible if and only if α is a regular family, when Y is an equiconnected topological space. In the case when X is a topological monoid, we study the relationship between the admissibility of the set open topology on C(X, Y ) and the continuity of an action of X on C(X, Y ). We prove that the set open topology on C(X, Y ) is admissible if and only if Cα (X, Y ) with the action s.f = (t → f (st)) is the cofree X-space over Y . © 2019 Elsevier B.V. All rights reserved.
1. Introduction Given two topological spaces X and Y we denote by C(X, Y ) the set of all continuous functions from X to Y . Let α be a family of subsets of X and Cα (X, Y ) be the set C(X, Y ) endowed with the set open topology. If τ is a topology on C(X, Y ) then the corresponding space is denoted by Cτ (X, Y ). A topology τ on the set C(X, Y ) is called admissible or jointly continuous if the evaluation mapping e : Cτ (X, Y ) × X → Y defined by e(f, x) = f (x) is continuous or equivalently, for every space Z, the continuity of a map f : Z −→ Cτ (X, Y ) implies that of the map fˆ : Z ×X −→ Y defined by fˆ(z, x) = f (z)(x) for every (z, x) ∈ Z × X. This notion was first discussed by Arens in [2] who proved that the compact open topology on C(X, Y ) is admissible when X is a regular locally compact space, and beyond that, this one is the least admissible topology on C(X, Y ). Afterwards, during their studies of splitting and admissible topologies on the spaces of continuous functions, Arens and Dugundji [3] showed that the uniform convergence topology
* Corresponding author. E-mail addresses: [email protected] (Abderrahmane Bouchair), [email protected] (Imane Dekkar). https://doi.org/10.1016/j.topol.2019.02.008 0166-8641/© 2019 Elsevier B.V. All rights reserved.
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is always admissible, they introduced also a class of topologies on C(X, Y ) by the help of a family α of subsets of X, called the set open topology, and they gave a sufficient condition for these topologies to be admissible. Since that time, many works have been focussed on the problem of the existence of admissible topologies on C(X, Y ). However, few of them provide necessary and sufficient conditions for the admissibility of these topologies, see for instance [10–12,17,18]. In the same context, Nokhrin [18] managed to prove that if the family α consists of closed subsets of X, then the set open topology on C(X, R) is admissible if and only if the family α is a regular family. Further properties concerning set open topologies are included in the papers [5], [13], [18] and [19]. Recently, Khosravi in [14] studied, for a topological monoid M , the cofree and injective M -spaces and M -flows over a completely regular (Tychonoff) space in the category, M -Tych, of all completely regular M -spaces with continuous M -morphisms. He established a relationship between the existence of the cofree object in the category M -Tych and the fact that the compact open topology on C(M, Y ) being admissible. It is worth mentioning that cofree objects, like all the other universal objects, are very important in categories because of their special property and thereby the question of the existence of such kind of objects in a category has been the main topic of several researches during the last decades. We refer the interested reader in this direction to [16]. Relying on the aforementioned works, our aim in the present paper is twofold. Firstly, we will give a necessary and sufficient condition for the admissibility of the set open topology on continuous function spaces in the case when the range space is an equiconnected space. Secondly, we will establish the interrelation between the admissibility of this last and the continuity of the actions of topological monoids on function spaces and obtain, by this, other necessary and sufficient conditions for the admissibility of this topology on C(M, Y ), in the case when M is a topological monoid. This paper is organized as follows. Section 2 contains notations and definitions that will be used in the paper. In Section 3, we prove that if α is a functional refinement family consisting of Y -compact subsets of a space X and Y is an equiconnected topological space with a base B consisting of Ψ-convex sets, then the set open topology on C(X, Y ) is admissible if and only if α is a regular family. In Section 4, we prove that if α is a compact network on a topological monoid M and Y is a topological space, then the set open topology on C(M, Y ) is admissible if and only if Cα (M, Y ) with the action s.f := (t → f (st)) is the cofree M -space over Y . In the last section, we study the existence of the cofree M -flow over Y , where M is a completely regular topological monoid and Y is a compact space. 2. Preliminaries Throughout this paper, X and Y are topological spaces and C(X, Y ) is the set of all continuous functions from X to Y . Let α be a nonempty family of subsets of X. The set open topology τα on C(X, Y ) has a subbase consisting of sets of the form [A, V ] = {f ∈ C (X) : f (A) ⊆ V }, where A ∈ α and V is an open subset of Y , and the set C(X, Y ) equipped with the set open topology is denoted by Cα (X, Y ). If V is not arbitrary but is restricted to some collection B of open subsets of Y , then we denote by CαB (X, Y ) the corresponding function space. Note that if α is replaced by the family of all finite unions of its elements, then the topology of Cα (X, Y ) does not change; so we can assume that the family α is closed with respect to finite unions. The symbols ∅ and N will stand for the empty set and the positive integers, respectively. We denote by R the real numbers with the usual topology. The interior and the closure of a subset A in X is denoted by intA and A, respectively. βX denotes the Čech–Stone compactification of the space X. A family α of subsets of X is called a network on X if for any point x ∈ X and any neighborhood U of x there is an A ∈ α such that x ∈ A ⊂ U . A compact network on X is a network in which each member of it is compact.
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It is proved in [17] that if α is a compact network on a topological space and Y is a completely regular space, then Cα (X, Y ) is a completely regular space. A family α of subsets of X is called a regular family [3] if for any point x ∈ X and for any neighborhood V of x, there exists an A ∈ α contained in V and containing x in its interior. Let X, Y and Z be topological spaces. The exponential function Λ : Y Z×X −→ (Y X )Z is defined by {[Λ(f )](z)}(x) = f (z, x) for each f : Z × X −→ Y , for each z ∈ Z and each x ∈ X. A topology τ on the set C(X, Y ) is called splitting or proper if for any space Z, the continuity of a map f : Z × X → Y implies that of the map Λ(f ) : Z → Cτ (X, Y ). Arens and Dugundji [3] proved that the compact open topology on C(X, Y ) is always splitting and if, in addition, X is a regular locally compact, then it coincides with the greatest splitting topology which always exists. It is also proved that if α is a family consisting of compact subsets of X, then the set open topology τα on C(X, Y ) is splitting. A topological space Y is said to be equiconnected [9] if there exists a continuous map Ψ : Y ×Y ×[0, 1] → Y such that Ψ(x, y, 0) = x, Ψ(x, y, 1) = y, and Ψ(x, x, t) = x for all x, y ∈ Y and t ∈ [0, 1]. The map Ψ is called an equiconnecting function. A subset V of an equiconnected space Y is called a Ψ-convex subset of Y if Ψ(V, V, [0, 1]) ⊆ V . It is a known fact that any topological vector space or any convex subset of any topological vector space is an equiconnected space and any equiconnected space is a pathwise connected space. Let M be a monoid and A be an arbitrary set. Recall that a set A is a (right) M -set if there is a right action λ : A × M −→ A such that a · s = λ(a, s), (a · s) · t = a · (st) and a · 1 = a. Let A and B be two M -sets. An M -homomorphism from A to B is a map f : A → B such that f (a · s) = f (a) · s, for each a ∈ A and for each s ∈ M . Let M be a monoid and A be an M -set. For any s ∈ M we define the M -homomorphism μs : A → A by μs (y) = y · s. We define also, for every a ∈ A, the M -homomorphism ρa : M → A by ρa (t) = a · t. In the (M ) (M ) case when A = M , we use the notation μs : M −→ M and ρs : M −→ M . Let M be a monoid endowed with a topology τM . M is called semitopological monoid if the multiplication (M ) (M ) mapping M × M −→ M is separately continuous. That is, the M -homomorphisms μs and ρs are continuous for every s ∈ M . If the multiplication mapping M × M −→ M is continuous, then M is called topological monoid. Let M be a topological monoid. An M -space is an M -set A with a topology τA such that the action A × M −→ A is continuous. Any compact M -space is called an M -flow. Let C be a concrete category over D and | , | : C −→ D be the forgetful functor. Let D be an object in D, A an object in C and ψ : |A| −→ D a morphism (in D). We say that A is the cofree object over D if for every object B in C and for every morphism h : |B| −→ D (in D), there exists a unique morphism h : B −→ A h| = h. Recall that in M -Act, the category of all M -sets and M -homomorphisms, in C such that ψ ◦ | the cofree M -set over a set Y is the set Y M of all mappings from M to Y with the action λ defined by λ(f, s) = f · s := (s −→ (f · s)(s ) = f (ss )), for every f ∈ Y M , and every s, s ∈ M . It is proved in [15] that in the category M -semiTop, where the objects are all semi-topological M -sets, the cofree object is Cp (M, Y ), the space C(M, Y ) equipped with the pointwise convergence topology. For standard terminology we refer to [8]. 3. Admissibility of set open topology on C(X,Y) for equiconnected space Y Throughout this section, X is a completely regular space. We will study the continuity and separate continuity of the evaluation mapping e : CαB (X, Y ) × X → Y when the range space Y is an equiconnected topological space. Recall that if x ∈ X, then the evaluation mapping at x is the map ex : C(X, Y ) → Y defined by ex (f ) = e(f, x) = f (x) for every f ∈ C(X, Y ). So, the separate continuity of the evaluation mapping e is equivalent to the continuity of all maps ex . It is clear that ex : Cp (X, Y ) → Y is continuous
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and so it is for any topology stronger than the topology of pointwise convergence on C(X, Y ). Therefore by Theorem 1.1.3 of [17], we have the following result. Corollary 3.1. Let X and Y be two topological spaces and α a family of subsets of X. If α is a network on X, then the evaluation mapping e : Cα (X, Y ) × X → Y is separately continuous. For the converse of the above corollary, we have the following theorem. Theorem 3.1. Let X be a topological space, α a family of closed subsets of X, and let Y be an equiconnected space with a base B consisting of Ψ-convex sets. The evaluation map e : CαB (X, Y ) × X → Y is separately continuous if and only if α is a network. Proof. Let x0 ∈ X and U an open neighborhood of it such that no F ∈ α containing x0 is contained in U . Let f : X → [0, 1] be a continuous function with f (x0 ) = 1 and f (X \ U ) = {0}. Take a nontrivial path p in Y with p(0) = p(1), an open basic set W ∈ B such that p(1) ∈ W ⊆ Y \ {p(0)}, and put g = p ◦ f . Then n g ∈ e−1 [Ai , Vi ], where Ai ∈ α and x0 (W ). Since ex0 is continuous, there exists an open neighborhood Og = i=1
Vi ∈ B for all i, of g such that Og ⊆ e−1 / Ai }. We have then I = ∅. If we suppose that x0 (W ). Put I = {i : x0 ∈ x0 belongs to Ai for each i ∈ {1, . . . , n}, then the continuous function h constantly equal to p(0) belongs to Og because p(0) ∈ Vi for each i from the fact that Ai U . But h does not belong to e−1 x0 (W ), which is a contradiction. So I = ∅. Then the set ∪i∈I Ai is closed and does not contain x0 . Since X is a completely regular space, there exists a continuous function ϕ : X → [0, 1] such that ϕ(x0 ) = 0 and ϕ( Ai ) = {1}. i∈I
Consider the function h1 : X → Y defined by h1 (t) = Ψ(p(0), g(t), ϕ(t)), for every t ∈ X. So h1 is continuous and belongs to Og . To see this last fact, take an arbitrary i ∈ {1, . . . , n} and take any point t ∈ Ai . If i ∈ I, then h1 (t) = g(t) ∈ Vi . If i ∈ / I, we get Ai ∩ (X \ U ) = ∅. Hence p(0) ∈ Vi . Ψ-convexity of Vi leads us to i∈I /
the fact that h1 (t) ∈ Vi . Whence h1 ∈ Og . On the other hand, we have h1 (x0 ) = p(0) ∈ / W . So h1 ∈ / e−1 x0 (W ) which contradicts the fact that ex0 is continuous. Hence, α is a network. 2 Recall that a family α of subsets of X is called a functional refinement [13] if for every A ∈ α, every finite n
sequence U1 , . . . , Un of open subsets of Y and every f ∈ C(X, Y ) such that A ⊆ ∪ f −1 (Ui ), there exists a i=1
finite sequence A1 , . . . , Am of members of α that refines f −1 (U1 ), . . . , f −1 (Un ) whose union contains A. A subset B of X is said to be Y -compact if for any continuous function f ∈ C(X, Y ), the set f (B) is compact in Y . Below, we give some well known properties of Y -compact sets. (1) For a Hausdorff space Y , any compact set is Y -compact in any space X. (2) There are Y -compact sets of a compact space that are not closed, see [19, Example 3.9]. (3) There are closed sets that are not Y -compact, see [19, Example 3.12]. It is proved [13] that if α is a functional refinement family consisting of Y -compact subsets of X and B is any arbitrary base for Y , then the family {[A, V ] : A ∈ α, V ∈ B} forms a subbase for the set open topology on C(X, Y ). That is, Cα (X, Y ) = CαB (X, Y ). So we have the following. Corollary 3.2. Let X be a topological space, α a functional refinement family consisting of closed Y -compact subsets of X and let Y be an equiconnected space with a base B consisting of Ψ-convex sets. Then the evaluation map e : Cα (X, Y ) × X → Y is separately continuous if and only if α is a network.
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To see under which conditions the set open topology on C(X, Y ) is admissible, we state the following result obtained by Arens and Dugundji in [3]. Theorem 3.2. [3, Theorem 4.7] Let X and Y be two topological spaces and α a family of subsets of X. If α is a regular family, then the set open topology τα on C(X, Y ) is an admissible topology. The converse of Theorem 3.2 is not true in general. Indeed, it is proved in [18, Example 3.8], that if X = [0, 1] and α is the family consisting of the point x = 0, all intervals [a, b] where 0 < a < b ≤ 1, and all sets of the form [0, a] ∪ {1}, then α is not a regular family. However the evaluation mapping e : Cα (X) × R → R is continuous. Note that, without any loss of generality, we assume that α contains finite unions of its members. Theorem 3.3. Let α be a family of closed subsets of X, and Y an equiconnected T1 -space with a base B consisting of Ψ-convex sets. Then the evaluation map e : CαB (X, Y ) × X → Y is continuous if and only if α is a regular family. Proof. We prove only the necessary condition. We will show that α is a regular family. To do this, let x0 be a point in X and U an open neighborhood of it. Since X is a completely regular space, there exists a continuous function f : X → [0, 1] such that f (x0 ) = 1 and f (X \ U ) = {0}. Take a path p in Y with p(0) = p(1), an open basic set W ∈ B such that p(1) ∈ W and p(0) ∈ / W , and put g = p ◦ f . Since the mapping e is continuous, there exists an open neighborhood V of x0 in X and an open neighborhood Og of g in CαB (X, Y ) such that e(Og × V ) ⊆ W . We have then V ⊆ U . Suppose, on the contrary, that there is x ∈ V \ U . Then e(g, x ) = g(x ) = p(0) ∈ / W , a contradiction. Therefore V ⊆ U . Let us take n Og = [Ai , Vi ], where Ai ∈ α and Vi ∈ B and put A = ∪{Ai : 1 ≤ i ≤ n and Ai ⊆ U }. We have then i=1
A = ∅. Indeed, suppose that Ai does not contain in U for each i. It follows that p(0) ∈ Vi , for each 1 ≤ i ≤ n. On the other hand, the continuous function h : X → Y defined by h(z) = p(0), for every z ∈ X, belongs to Og . But e(h, V ) = h(V ) ⊆ {p(0)} ⊂ W , which is a contradiction. Hence A = ∅. Now, let us prove that x0 ∈ intA. Suppose, on the contrary, that x0 ∈ / intA. As above, it is easy to check that there is at least i0 for which x0 ∈ / Ai0 . Let us put V = V ∩ (X \ ∪{Ai : x0 ∈ / Ai }). Clearly, V is an open neighborhood of x0 which is not contained in A. Take a point x1 ∈ V \ A and put I = {i : x1 ∈ / Ai }. Since X is a completely regular space, there is a continuous function ϕ : X → [0, 1] such that ϕ(x1 ) = 0 and ϕ( Ai ) = {1}. Then i∈I
the function h1 : X → Y defined by h1 (t) = Ψ(p(0), g(t), ϕ(t)), for every t ∈ X, is continuous and belongs to Og . Indeed, let i ∈ {1, . . . , n} and take any point t ∈ Ai . If i ∈ I, then h1 (t) = g(t) ∈ Vi . If i ∈ / I, since x1 ∈ Ai we have x0 ∈ Ai and Ai does not contain in A. So Ai ∩(X \U ) = ∅. Hence p(0) ∈ Vi . Ψ-convexity i∈I /
of Vi leads us to the fact that h1 (t) ∈ Vi . Whence h1 ∈ Og . On the other hand, we have e(h1 , x1 ) = p(0). This contradiction proves that x0 ∈ intA. Thus, the family α is a regular family. 2 Corollary 3.3. Let α be a functional refinement family consisting of closed Y -compact subsets of a topological space X and Y be an equiconnected T1 -space with a base B consisting of Ψ-convex sets. Then, the set open topology τα on C(X, Y ) is admissible if and only if α is a regular family. 4. Admissibility of set open topologies and cofree M -spaces over a topological space Let M be a topological monoid and Y a topological space. In this section, we present the relationship between the fact that set open topology on the function space C(M, Y ) being admissible and the continuity of the action of M on C(M, Y ).
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Theorem 4.1. Let M be a topological monoid, Y a topological space, and α be a network consisting of compact subsets of M . The following statements are equivalent. 1. The set open topology τα on C(M, Y ) is admissible. 2. Cα (M, Y ) with the action λ defined by λ(f, s) = f · s := (s −→ (f · s)(s ) = f (ss )), for every f ∈ Cα (M, Y ) and s, s ∈ M , is the cofree M -space over Y . Proof. (1) ⇒ (2) We will first prove that Cα (M, Y ) with the action λ mentioned above is an M -space. Afterwards, we shall show that is cofree over the space Y . We have that λ is an action. In fact, let f ∈ Cα (M, Y ) and s1 , s2 ∈ M . We have that for every s ∈ M [(f · s1 ) · s2 ](s ) = (f · s1 )(s2 s ) = f (s1 s2 s ) = [f · (s1 s2 )](s ), so (f · s1 ) · s2 = f · (s1 s2 ). Furthermore, we have (f · 1M )(s ) = f (1M s ) = f (s ), for every s ∈ M . Then f · 1M = f . Now let us prove that λ is continuous. Let f ∈ Cα (M, Y ), s ∈ M , B ∈ α and U be an open subset in Y such that f · s ∈ [B, U ]. We have f · s ∈ [B, U ] ⇔ ∀b ∈ B, (f · s)(b) ∈ U ⇔ ∀b ∈ B, e(f, sb) ∈ U. Since the set open topology on C(M, Y ) is admissible, the evaluation map e is continuous. Then for every b ∈ B there exist an open subset Wsb in M which contains sb and an open neighborhood Of of f in Cα (M, Y ) such that e(Of × Wsb ) ⊆ U.
(1)
Furthermore, since M is a topological monoid, there exist two open neighborhoods Wsb and Wb of s and b respectively, such that Wsb Wb ⊆ Wsb . n Since B ⊆ b∈B Wb and B is compact, there exists {b1 , b2 , . . . , bn } ⊂ B such that B ⊆ i=1 Wbi . Put n Ws = i=1 Wsbi , which is open. We have then Ws B ⊆
n
Wsbi
i=1
n
Wbi ⊆
i=1
n
Wsbi Wbi ⊆
i=1
n
Wsbi .
i=1
From (1) we get e(Of ×
n i=1
Wsbi ) =
n
e(Of × Wsbi ) ⊆ U.
i=1
Then e(Of × Ws B) ⊆ e(Of ×
n
Wsbi ) ⊆ U.
(2)
i=1
We have that Of × Ws is open in Cα (M, Y ) × M which contains (f, s). We will prove in the sequel that λ(Of × Ws ) = Of · Ws ⊆ [B, U ]. Let g ∈ Of · Ws . Then there exist f0 ∈ Of and s0 ∈ Ws such that g = f0 · s0 . Thus, for every b ∈ B, we obtain
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g(b) = (f0 · s0 )(b) = f0 (s0 b) = e(f0 , s0 b) ∈ e(Of × Ws B). From (2) we have g(b) ∈ U , and so g(B) ⊂ U . Therefore g ∈ [B, U ]. Thus, the action λ is continuous and hence Cα (M, Y ) is an M -space. It remains to show that is cofree over Y . Consider the map ψ : Cα (M, Y ) −→ Y defined by ψ(f ) = f (1M ), for every f ∈ C(M, Y ). This map is continuous. In fact, let f ∈ Cα (M, Y ) and U an open neighborhood of ψ(f ). We have 1M ∈ f −1 (U ). Since α is a network, there exists B0 ∈ α such that 1M ∈ B0 ⊂ f −1 (U ). We set W = [B0 , U ]. Then f ∈ W and ψ(W ) ⊂ U . This proves that ψ is continuous. Take an M -space A and a continuous map h : A −→ Y , and let us show that there exists a unique continuous M -homomorphism h = h. Consider the map h : A −→ Cα (M, Y ) such that ψ ◦ h from A to Cα (M, Y ) defined by h(a))(s ) = h(a · s )) h(a) := (s −→ ( for every a ∈ A and s ∈ M . Thus, for every s, s ∈ M , we have [ h(a) · s](s ) = [ h(a)](ss ) = h(a · (ss )) = h((a · s) · s )) = [ h(a · s)](s ). So h is an M -homomorphism. If we denote by λ1 the action of M over A, then the map h ◦ λ1 :
A×M (a, s)
−→ A −→ Y −→ a · s −→ h(a · s)
is continuous. Since the set open topology in this case is splitting because the family α consists of compact sets, then Λ(h ◦ λ1 ) is continuous. Furthermore, we have that for every a ∈ A and s ∈ M {[Λ(h ◦ λ1 )](a)}(s) = (h ◦ λ1 )(a, s) = h(λ1 (a, s)) = h(a · s) = [ h(a)](s). Hence Λ(h ◦ λ1 ) = h. This allows us to deduce the continuity of h. Besides, (ψ ◦ h)(a) = ψ( h(a)) = ( h(a))(1M ) = h(a · 1M ) = h(a), for every a ∈ A. Then ψ ◦ h = h. Now, suppose that there exists g : A −→ Cα (M, Y ) an M -homomorphism such that ψ ◦ g = h. We have then that for every a ∈ A and s ∈ M (g(a))(s) = (g(a))(s · 1M ) = (g(a) · s)(1M ) = ψ(g(a) · s) = (ψ ◦ g)(a · s) = h(a · s) = ( h(a))(s). Hence g = h, and so h is unique. (2) ⇒ (1) Since Cα (M, Y ) is an M -space and the map ψ : Cα (M, Y ) −→ Y defined above is continuous, the map ψ ◦ λ is continuous. Furthermore, for every (f, s) in Cα (M, Y ) × M we have (ψ ◦ λ)(f, s) = ψ(f · s) = (f · s)(1M ) = f (s) = e(f, s). That is, e = ψ ◦ λ. Thus, the evaluation mapping e is continuous which means that the set open topology on C(M, Y ) is admissible. 2
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Recall that a topological space is called an Alexandroff space [1] if the intersection of any family of open subsets of it is open. We have the following theorem. Theorem 4.2. Let M be an Alexandroff topological monoid, Y a topological space and α be a network on M . Then the following statements are equivalent. 1. The set open topology τα on C(M, Y ) is admissible. 2. Cα (M, Y ) with the action λ defined by λ(f, s) = f · s := (s −→ (f · s)(s ) = f (ss )), for every f ∈ Cα (M, Y ) and s, s ∈ M , is an M -space. Proof. (1) ⇒ (2) As in the proof of Theorem 4.1, λ is an action. For the continuity of λ we follow the same argument of this proof by taking Ws = b∈B Wsb , which is an open subset of M because it is an Alexandroff space. Hence Cα (M, Y ) is an M -space. (2) ⇒ (1) The map ψ : Cα (M, Y ) −→ Y defined by ψ(f ) = f (1M ), for every f ∈ C(M, Y ), is continuous with the same reasoning as in the proof of Theorem 4.1. Since Cα (M, Y ) with the action λ is an M -space, the map e = ψ ◦ λ is continuous. Hence the set open topology τα on C(M, Y ) is admissible. 2 Remark 4.1. It is useful to mention that we can not confirm here that Cα(M, Y ) is cofree over Y as in Theorem 4.1, this is because the set open topology in this case is not splitting. 5. Admissibility of set open topologies and cofree M -flows over a topological space Using Theorem 4.1 above we study, in this section, the existence of the cofree M -flow over Y , where M is a completely regular topological monoid and Y is a compact space. We denote by βCα (M, Y ) the Čech–Stone compactification of the function space Cα (M, Y ) and by β(Cα (M, Y ) × M ) the Čech–Stone compactification of the product space Cα (M, Y ) × M . Theorem 5.1. Let M be a completely regular topological monoid, Y a compact space and α be a compact network on M . The following statements are equivalent. 1. The set open topology τα on C(M, Y ) is admissible. 2. There exists an action : βCα (M, Y ) × M −→ βCα (M, Y ), λ |C (M,Y )×M coincides with the action of Cα (M, Y ) and βCα (M, Y ) is the cofree M -flow such that λ α over Y . Proof. (1) ⇒ (2) Suppose that the set open topology on C(M, Y ) is admissible. Then, by Theorem 4.1, then we shall prove that Cα (M, Y ) with the action λ is the cofree M -space over Y . Firstly we will define λ, it is continuous and finally we will show that βCα (M, Y ) is cofree over Y . Since α is a compact network and Y is compact, then Cα (M, Y ) is completely regular. This leads to the fact that β(Cα (M, Y ) × M ) exists and for the action λ : Cα (M, Y ) × M −→ Cα (M, Y ) there exists, by Corollary 3.6.6 in [8], a continuous ¯ : β(Cα (M, Y ) × M ) −→ βCα (M, Y ) such that λ ¯ |C (M,Y )×M = λ. map λ α Fix an arbitrary element t ∈ M and define the mapping k : Cα (M, Y ) −→ Cα (M, Y ) × M , such as k(f ) = (f, t), ∀f ∈ Cα (M, Y ).
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It is clear that this map is continuous and we have k(Cα (M, Y )) = {(f, t) : f ∈ Cα (M, Y )} = Cα (M, Y ) × {t}. By definition of compactification, k(Cα (M, Y )) ⊂ Cα (M, Y ) × M can be considered as a subspace of β(Cα (M, Y ) × M ) and, so, the closure of k(Cα (M, Y )) will be compact in β(Cα (M, Y ) × M ). Furthermore, there exists a compact space B such that the closure of k(Cα (M, Y )) in β(Cα (M, Y ) × M ) is equal to B × {t}. Let us denote by ϕ0 the projection ϕ0 : B × M −→ B. Using Corollary 3.6.6 from [8] once again, the mapping ϕ0 ◦k can be extended to a continuous mapping h : βCα (M, Y ) −→ B that makes the following diagram commutative β
−→ βCα (M, Y ) ↓h −→ B
Cα (M, Y ) k ↓ B×M
ϕ0
Put λ = λ|B×M and define := λ ◦ (h × idM ) : βCα (M, Y ) × M −→ B × M −→ βCα (M, Y ). λ is continuous and we have also that for each s ∈ M and every f ∈ Cα (M, Y ), Clearly λ ¯ |B×M (h(f ), s) = λ ¯ |B×M [(ϕ0 (f, t), s)] = λ ¯ |B×M (f, s) = λ(f, s). s) = [λ ¯ |B×M ◦ (h × idM )](f, s) = λ λ(f, |C (M,Y )×M = λ. Now, let us prove that λ is an action. Let g ∈ βCα (M, Y ) and s, s ∈ This means that λ α M . The space Cα (M, Y ) can be considered as a dense subspace in βCα (M, Y ). Then there exists a net (fi )i∈I ⊂ Cα (M, Y ) converging to g. We have λ(lim λ(g, s), s ). ss ) = lim λ(f i , ss ) = lim λ(fi , ss ) = lim λ( λ(f i , s), s ) = λ( fi , s), s ) = λ( λ(g, i
i
i
i
is an action. We deduce that βCα (M, Y ) is an M -space. Since it is compact, it is an M -flow. It Hence λ remains to show that βCα (M, Y ) is cofree over Y . The continuous function ψ : Cα (M, Y ) → Y defined by ψ(f ) = f (1M ), for every f ∈ Cα (M, Y ), can be extended to a continuous function ψ¯ : βCα (M, Y ) −→ Y . Let us take an M -flow F and a continuous function h : F −→ Y . Since Cα (M, Y ) is cofree over Y , there exists an unique continuous M -homomorphism h = h. Since h : F −→ Cα (M, Y ) such that ψ ◦ ψ¯|Cα (M,Y ) = ψ, we have that for every y ∈ F ¯ (ψ¯ ◦ h)(y) = ψ[ h(y)] = ψ[ h(y)] = (ψ ◦ h)(y) = h(y). on βCα (M, Y ) where βCα (M, Y ) with this action (2) ⇒ (1) Suppose that there exists a continuous action λ is an M -flow and λ|Cα (M,Y )×M = λ, with λ is the action of M on Cα (M, Y ). We have then Cα (M, Y ) is an M -space, and since ψ : Cα (M, Y ) −→ Y is continuous, then the mapping λ
ψ
ψ ◦ λ : Cα (M, Y ) × M −→ Cα (M, Y ) −→ Y is also continuous. Furthermore, we have for every f ∈ Cα (M, Y ) and s ∈ M (ψ ◦ λ)(f, s) = ψ(λ(f, s)) = ψ(f · s) = (f · s)(1M ) = f (s) = e(f, s). This means that e = ψ ◦ λ which implies that the evaluation mapping e is continuous. Hence the set open topology on Cα (M, Y ) is admissible. 2
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In the end, it is interesting to discuss the injectivity of cofree objects, in case of existence, in the category M -Tych and the one of M -flows. It is known that injectivity of objects is a very useful notion which category theory inherited from homological and commutative algebra [20], and it has been studied, significantly, in different categories such as the category Pos of partial ordered sets and monotone mappings, the category Top of topological spaces, the category Boo of boolean algebras and boolean homomorphisms . . . etc., see for example [4], [6], [7] and the references cited therein. Let us take a completely regular topological monoid M and a completely regular space Y . If α is a regular family of compact subsets of M , then the set open topology on C(M, Y ) is admissible and splitting [3]. Moreover, M is a regular locally compact space in this case and thus the compact open topology is admissible too and it coincides with the set open topology, i.e. Cα (M, Y ) = Ck (M, Y ), where Ck (M, Y ) denotes the set of continuous functions endowed with the compact open topology. Consequently, Ck (M, Y ) is the cofree M -space over Y . Here we note that Ck (M, Y ) is completely regular, and it is injective in the category M -Tych if and only if Y is injective in the category Tych (Proposition 3.3 of [14]). As well, Ck (M, Y ) is injective in the category Tych if and only if Ck (M, Y ) is injective in the category M-Tych (Proposition 3.5 of [14]). If in addition Y is a compact space, then the cofree M -flow over Y , βCk (M, Y ), is injective in the category of M -flows if and only if Y is injective in the category of compact spaces (Proposition 3.4 of [14]). Acknowledgements The authors would like to thank the anonymous referee for his/her careful reading and insightful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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