- Email: [email protected]
Projectively first-countable semitopological groups
Topology and its Applications 204 (2016) 246–252
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Projectively first-countable semitopological groups Iván Sánchez Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Campus de Riu Sec s/n, Castelló, Spain
a r t i c l e
i n f o
Article history: Received 19 December 2014 Received in revised form 16 March 2016 Accepted 17 March 2016
a b s t r a c t We show that a regular semitopological group G is projectively regular firstcountable if and only if G is ω-balanced, locally ω-good and Ir(G) ≤ ω. Similarly, we give an internal characterization of projectively Ti first-countable semitopological groups, for i = 0, 1, 2. © 2016 Elsevier B.V. All rights reserved.
MSC: 54H11 54A25 54B10 54C25 54D10 Keywords: Semitopological groups Projectively first-countable Index of regularity Hausdorff number Symmetry number ω-balanced Locally ω-good
1. Introduction In 1953, G.I. Katz showed that a Hausdorff topological group G is topologically isomorphic to a subgroup of a product of first-countable (metrizable) topological groups if and only if G is ω-balanced (see [4]). As in [7], in this paper, a semitopological group G is regular if it is Hausdorff and for every open neighborhood U of the identity e in G, there exists an open neighborhood V of e such that V ⊆ U (usually this kind of spaces are called T3 ). Let P be a topological property. In [7], M. Tkachenko defines when a paratopological (topological) group G is projectively P. Similarly, we say that a semitopological group G is projectively P if for every neighborhood U of the identity in G, there exists a continuous homomorphism p : G → H onto a semitopological group
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.03.017 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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H with property P such that p−1 (V ) ⊆ U , for some neighborhood V of the neutral element in H. It is easy to see that a (regular, Tychonoff) Ti semitopological group is projectively (regular, Tychonoff) Ti first-countable if and only if it admits a homeomorphic embedding as a subgroup into a product of (regular, Tychonoff) Ti first-countable semitopological groups, for i = 0, 1, 2. M. Tkachenko extended Katz’s Theorem to the class of regular paratopological groups showing that a regular paratopological group G can be embedded as a subgroup of a product of regular first-countable paratopological groups if and only if G is ω-balanced and Ir(G) ≤ ω. In Theorem 2.9, we show that a regular semitopological group G is projectively regular first-countable if and only if G is ω-balanced, locally ω-good and Ir(G) ≤ ω. Similarly, in Theorems 2.11, 2.12 and 2.13, we give an internal characterization of projectively Ti first-countable semitopological groups, for i = 0, 1, 2. In [1], the reader can find the basic notions of semitopological (paratopological) groups. 2. Projectively first-countable semitopological groups Let G be a semitopological group with identity e. A subset V of G is called ω-good if there exists a countable family γ ⊂ N (e) such that for every x ∈ V , we can find W ∈ γ with xW ⊆ V . Denote by N ∗ (e) the family of ω-good sets of G which contain the identity. A semitopological group G with identity e is called locally ω-good if the family N ∗ (e) is a local base at e in G. According to the following result, every paratopological group is locally ω-good. Lemma 2.1. ([6, Lemma 3.10]) Every paratopological group G has a local base at the neutral element consisting of ω-good sets. The previous result cannot be extended to the class of T1 semitopological groups. Example 2.2. There exists a T1 compact semitopological group which is not locally ω-good. Proof. Consider the additive group R of real numbers. Let τ be the co-finite topology on R, i.e., τ = {U ⊆ R : |R \ U | < ω} ∪ {∅}. Clearly, G = (R, τ ) is T1 compact semitopological group. Let us show that G is not locally ω-good. Suppose the contrary that G is locally ω-good. Put U = G \ {1}. Then we can find V ∈ N ∗ (e) such that V ⊆ U . Clearly V is uncountable. Since V ∈ N ∗ (e), there exists a countable family γ = {Un ∈ N (e) : n ∈ ω} such that for every x ∈ V , we can choose n ∈ ω with xUn ⊆ V . We can assume that for each n ∈ ω, Un = G \ Fn , where Fn is a finite subset of G. Put F = n∈ω Fn . By the definition of τ , we have that V \ F = G \ A, where A is a countable subset of G. Hence there exists a ∈ G such that a, 1 − a ∈ V \ F . It follows that 1 = a + (1 − a) ∈ a + Un for every n ∈ ω. This contradiction shows that G is not locally ω-good. 2 In the following two propositions, we show that the property “locally ω-good” is closed with respect to subgroups and topological products. Proposition 2.3. Let {Gi : i ∈ I} be a family of locally ω-good semitopological groups. Then the product G = i∈I Gi is locally ω-good. Proof. Let U be an open neighborhood of the identity e in G. We can assume that U is a canonical open set, with U = i∈I Ui and Ui = Gi for i ∈ {i1 , . . . , in }. Let ei be the identity in Gi , for every i ∈ I. Since every Gi is locally ω-good, for each 1 ≤ k ≤ n, we can find Vik ∈ N ∗ (eik ) such that Vik ⊆ Uik . Put V = i∈I Wi , where Wi = Gi if i ∈ / {i1 , . . . , in } and Wik = Vik for each 1 ≤ k ≤ n. It is easy to see that V ∈ N ∗ (e) and V ⊆ U . This finishes the proof. 2
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Proposition 2.4. If H is a subgroup of a locally ω-good semitopological group G, then H is locally ω-good. Proof. Let U be an open neighborhood of the identity e in G. By hypothesis, there exists V ∈ N ∗ (e) such that V ⊆ U . Since V ∈ N ∗ (e), we can find a countable family γ ⊆ N (e) such that for every x ∈ V , there exists W ∈ γ satisfying xW ⊆ V . Put γ = {H ∩ W : W ∈ γ}. Clearly, γ is countable. Let us show that V ∩ H is an ω-good set of H. Indeed, fix h ∈ H ∩ V . Take W ∈ γ satisfying hW ⊆ V . We have h(H ∩ W ) = H ∩ hW ⊆ H ∩ V. Since H ∩ V is an ω-good set of H and H ∩ V ⊆ H ∩ U , H is locally ω-good.
2
The next result is evident. Proposition 2.5. Every first-countable semitopological group is locally ω-good. Corollary 2.6. Every subgroup of an arbitrary product of first-countable semitopological groups is locally ω-good. Proof. Apply Propositions 2.3–2.5.
2
Given a semitopological group G with identity e, we say that a family γ ⊆ N (e) is subordinated to a neighborhood U of the identity in G if for each x ∈ G there exists V ∈ γ such that xV x−1 ⊆ U . The semitopological group G is ω-balanced if for every U ∈ N (e), there exists a countable family γ subordinated to U . The following lemma plays an important role in this paper. Lemma 2.7. Let G be a semitopological group with identity e. Suppose that a family γ ⊂ N (e) satisfies the following conditions: (a) for every U ∈ γ and x ∈ U , there exists V ∈ γ such that xV ⊂ U ; (b) γ is subordinated to U , for each U ∈ γ. Then the set N = U ∈ γ.
{U ∩ U −1 : U ∈ γ} is an invariant subgroup of G. Further, U N = N U = U for each
Proof. It is clear that N = N −1 . Let us show that N is a subgroup of G. Take a, b ∈ N and U ∈ γ. It follows that a, b ∈ U ∩U −1 . By (a), there exists V ∈ γ such that aV ⊆ U . Hence ab ∈ aN ⊆ aV ⊆ U . Since b−1 ∈ U , we can find W ∈ γ satisfying b−1 W ⊆ U , equivalently, W −1 b ⊆ U −1 . We have that ab ∈ N b ⊆ W −1 b ⊆ U −1 . It follows that ab ∈ U ∩ U −1 for each U ∈ γ. Therefore, ab ∈ N . We have thus proved that N is a subgroup of G. Now, let us show that N is invariant in G. Take x ∈ G and U ∈ γ. By (b), we can find V ∈ γ such that xV x−1 ⊆ U . Hence xN x−1 ⊆ xV x−1 ⊆ U and xN x−1 ∈ xV −1 x−1 ⊆ U −1 . This implies that xN x−1 ∈ U ∩ U −1 for every U ∈ γ. This shows that N is invariant in G. For the last part, pick up U ∈ γ. Take a ∈ U . By (a), we can find V ∈ γ such that aV ⊆ U . It follows that aN ⊆ aV ⊆ U . Therefore, U N = U for each U ∈ γ. Since N is invariant in G, uN = N u for each u ∈ U ∈ γ. It follows that N U = U N = U for every U ∈ γ. 2 As in [7], in this paper, a semitopological group G is regular if it is Hausdorff and for every open neighborhood U of the identity e in G, there exists an open neighborhood V of e such that V ⊆ U
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(usually this kind of spaces are called T3 ). The following cardinal function permits us to give an internal characterization of projectively regular first-countable semitopological groups. Definition 2.8. ([7]) Denote by N (e) the family of neighborhoods of the identity e in a semitopological group G. We recall that the index of regularity Ir(G) of a regular semitopological group G is the minimum cardinal number κ such that for every neighborhood U of the identity e in G, we can find V ∈ N (e) and a family γ ⊆ N (e) such that W ∈γ V W −1 ⊆ U and |γ| ≤ κ. Theorem 2.9. Let G be a regular semitopological group. Then G is projectively regular first-countable if and only if G is ω-balanced, locally ω-good and Ir(G) ≤ ω. Proof. Suppose that G is projectively regular first-countable, then G is topologically isomorphic to a sub group of Π = i∈I Hi , where Hi is a regular first-countable semitopological group for each i ∈ I. By [7, Corollary 3.4], G is ω-balanced and Ir(G) ≤ ω. By Corollary 2.6, G is locally ω-good. Now, suppose that G is an ω-balanced, locally ω-good and Ir(G) ≤ ω. Let e be the identity of G. Take an arbitrary U0 ∈ N (e). We shall define a continuous homomorphism p : G → H onto a regular first-countable semitopological group H such that p−1 (V0 ) ⊆ U0 for some V0 ∈ N (eH ). We will construct by induction a sequence {γn : n ∈ ω} satisfying the following conditions for each n ∈ ω: (i) (ii) (iii) (iv) (v) (vi)
γn ⊆ N ∗ (e) and |γn | ≤ ω; γn ⊆ γn+1 ; γn is closed under finite intersections; the family γn+1 is subordinated to U , for each U ∈ γn ; for every U ∈ γn and x ∈ U , there exists V ∈ γn+1 such that xV ⊆ U ; for each U ∈ γn , there exists V ∈ γn+1 such that W ∈γn+1 V W −1 ⊆ U .
Since G is locally ω-good, there exists U0∗ ∈ N ∗ (e) such that U0∗ ⊆ U0 . Put γ0 = {U0∗ }. Suppose that for some n ∈ ω, we have defined families γ0 , . . . , γn satisfying (i)–(vi). Since γn is countable and G is ω-balanced, we can find a countable family λn,1 ⊆ N ∗ (e) subordinated to every U ∈ γn . As γn ⊆ N ∗ (e), there exists a countable family λn,2 ⊆ N ∗ (e) such that for each U ∈ γn and x ∈ U , there exists V ∈ λn,2 satisfying xV ⊆ U . Since Ir(G) ≤ ω, we can find a countable family λn,3 ⊆ N ∗ (e) such that for every U ∈ γn , there exists V ∈ λn,3 satisfying W ∈λn,3 V W −1 ⊆ U . Let γn+1 be the minimal family containing 3 γn ∪ i=1 λn,i and closed under finite intersections. Clearly, γn+1 is countable and satisfies (i)–(vi). This finishes our construction. Note that γ = n∈ω γn is countable and satisfies conditions (a)–(b) of Lemma 2.7. Hence, the set N = {V ∩ V −1 : V ∈ γ} is an invariant subgroup of G. It follows from the construction of γ that −1 ⊆ U , for every U ∈ γ, so N = V ∈γ V V −1 . Consider the algebraic group H = G/N . Let V ∈γ V V p : G → H be the canonical homomorphism. Put B = {p(V ) : V ∈ γ}. We claim that the family B satisfies the following properties: (1) for each A, B ∈ B, there exists C ∈ B such that C ⊆ A ∩ B; (2) for all A ∈ B and a ∈ A, there exists B ∈ B such that aB ⊆ A; (3) B is subordinated to each A ∈ B. Indeed, since γ is closed under finite intersections, condition (1) holds. Conditions (2) and (3) follow from (iv)–(v). Conditions (1)–(3) imply that there exists a topology τ on H such that (H, τ ) is a semitopological group and the family B is a local base for H at eH . Since B is countable, (H, τ ) is first-countable.
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Let us show that the semitopological group (H, τ ) is regular. First, we prove that (H, τ ) is Hausdorff. Take y ∈ H \ {eH }, where eH is the identity of H. Choose x ∈ G such that p(x) = y. Since y = eH , we have that x ∈ / N = V ∈γ V V −1 . Therefore, there exists V ∈ γ such that x ∈ / V V −1 , equivalently, xV ∩ V = ∅. It follows from Lemma 2.7 that xV N ∩ V N = ∅, so yp(V ) ∩ p(V ) = ∅. Now, choose U ∈ γ. Then U ∈ γn for some n ∈ ω. By (vi), we can find V ∈ γn+1 such that −1 ⊆ U . Let us show that p(V ) ⊆ p(U ). Take y ∈ H \ p(U ) and choose x ∈ G such that W ∈γn+1 V W p(x) = y. Clearly x ∈ / U . Therefore, there exists W ∈ γn+1 such that x ∈ / V W −1 , equivalently, xW ∩ V = ∅. It follows from Lemma 2.7 that xW N ∩ V N = ∅, so yp(W ) ∩ p(V ) = ∅. We have thus proved that y ∈ / p(V ). We conclude that (H, τ ) is regular. Finally, V0 = p(U0∗ ) is an open neighborhood of eH in H. It follows from Lemma 2.7 that p−1 (V0 ) = ∗ U0 N = U0∗ ⊆ U0 . 2 In Theorem 2.11, we generalize Katz’s theorem to the class of Hausdorff semitopological groups. First, we need the next cardinal function. Definition 2.10. ([7]) For a Hausdorff semitopological group G with identity e, the Hausdorff number of G, denoted by Hs(G), is the minimum cardinal number κ such that for every neighborhood U of e in G, there exists a family γ of neighborhoods of e such that V ∈γ V V −1 ⊆ U and |γ| ≤ κ. Theorem 2.11. A Hausdorff semitopological group is projectively Hausdorff first-countable if and only if it is ω-balanced, locally ω-good and has countable Hausdorff number. Proof. Let G be a Hausdorff semitopological group. Suppose that G is projectively Hausdorff first-countable, then G is topologically isomorphic to a subgroup of Π = i∈I Hi , where Hi is a Hausdorff first-countable semitopological group for each i ∈ I. Clearly, G is ω-balanced and locally ω-good. By Propositions 2.1–2.3 in [7], we have that Hs(G) ≤ ω. Now, suppose that G is an ω-balanced, locally ω-good and Hs(G) ≤ ω. Let e be the identity of G. Take an arbitrary U0 ∈ N (e). We shall define a continuous homomorphism p : G → H onto a Hausdorff first-countable semitopological group H such that p−1 (V0 ) ⊆ U0 for some V0 ∈ N (eH ). We will construct by induction a sequence {γn : n ∈ ω} satisfying the following conditions for each n ∈ ω: (i) (ii) (iii) (iv) (v) (vi)
γn ⊆ N ∗ (e) and |γn | ≤ ω; γn ⊆ γn+1 ; γn is closed under finite intersections; the family γn+1 is subordinated to U , for each U ∈ γn ; for every U ∈ γn and x ∈ U , there exists V ∈ γn+1 such that xV ⊆ U ; −1 ⊆ U , for each U ∈ γk . V ∈γn+1 V V
Since G is locally ω-good, we can find U0∗ ∈ N ∗ (e) such that U0∗ ⊆ U0 . Put γ0 = {U0∗ }. Suppose that for some n ∈ ω, we have defined families γ0 , . . . , γn satisfying (i)–(vi). Since γn is countable and G is ω-balanced, we can find a countable family λn,1 ⊆ N ∗ (e) subordinated to every U ∈ γn . As γn ⊆ N ∗ (e), there exists a countable family λn,2 ⊆ N ∗ (e) such that for each U ∈ γn and x ∈ U , we can choose V ∈ λn,2 satisfying xV ⊆ U . Since Hs(G) ≤ ω, we can find a countable family λn,3 ⊆ N ∗ (e) such that V ∈λn,3 V V −1 ⊆ U for 3 every U ∈ γn . Let γn+1 be the minimal family containing γn ∪ i=1 λn,i and closed under finite intersections. Clearly, γn+1 is countable and satisfies (i)–(vi). This finishes our construction. Note that γ = n∈ω γn is countable and satisfies conditions (a)–(b) of Lemma 2.7. Hence, the set N = {V ∩ V −1 : V ∈ γ} is an invariant subgroup of G. It follows from the construction of γ that V ∈γ V V −1 ⊆ U , for every U ∈ γ, so N = V ∈γ V V −1 . Consider the algebraic group H = G/N . Let p : G → H be the
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canonical homomorphism. Put B = {p(V ) : V ∈ γ}. Arguing as in the proof of Theorem 2.9, we conclude that there exists a topology τ on H such that (H, τ ) is a first-countable semitopological group and the family B is a local base for H at eH . Let us show that the semitopological group (H, τ ) is Hausdorff. Take y ∈ H \ {eH }, where eH is the identity of H. Choose x ∈ G such that p(x) = y. Since y = eH , we have that x ∈ / N = V ∈γ V V −1 . Therefore, there exists V ∈ γ such that x ∈ / V V −1 , equivalently, xV ∩ V = ∅. It follows from Lemma 2.7 that xV N ∩ V N = ∅, so yp(V ) ∩ p(V ) = ∅. We have thus proved that (H, τ ) is Hausdorff. Finally, V0 = p(U0∗ ) is an open neighborhood of eH in H and p−1 (V0 ) = U0∗ N = U0∗ ⊆ U0 . 2 The symmetry number of a T1 semitopological group G, denoted by Sm(G), as the minimum cardinal number κ such that for every neighborhood U of e in G, there exists a family γ ⊆ N (e) such that −1 ⊆ U and |γ| ≤ κ (see [5]). V ∈γ V Theorem 2.12. A T1 semitopological group G is projectively T1 first-countable if and only if G is ω-balanced, locally ω-good and Sm(G) ≤ ω. Proof. Suppose that G is projectively T1 first-countable. Then G is ω-balanced G and locally ω-good. By propositions 2.1–2.3 in [5], we have that Sm(G) ≤ ω. Now, suppose that G is ω-balanced, locally ω-good and Sm(G) ≤ ω. Let e be the identity of G. Take an arbitrary U0 ∈ N (e). We shall define a continuous homomorphism p : G → H onto a T1 first-countable semitopological group H such that p−1 (V0 ) ⊆ U0 for some V0 ∈ N (eH ). As in the proofs of Theorems 2.9 and 2.11, we can construct a sequence {γn : n ∈ ω} satisfying the following conditions for each n ∈ ω: (i) (ii) (iii) (iv) (v) (vi) (vii)
γ0 = {U0∗ }; γn ⊆ N ∗ (e) and |γn | ≤ ω; γn ⊆ γn+1 ; γn is closed under finite intersections; the family γn+1 is subordinated to U , for each U ∈ γn ; for every U ∈ γn and x ∈ U , there exists V ∈ γn+1 such that xV ⊆ U ; −1 ⊆ U , for every U ∈ γn . V ∈γn+1 V
The family γ = n∈ω γn is countable and satisfies conditions (a)–(b) of Lemma 2.7. This implies that N = {V ∩ V −1 : V ∈ γ} is an invariant subgroup of G. It follows from the construction of γ that N = V ∈γ V = V ∈γ V −1 . Consider the algebraic group H = G/N . Let p : G → H be the canonical homomorphism. Put B = {p(V ) : V ∈ γ}. Arguing as in the proof of Theorem 2.9, we conclude that the family B satisfies the following conditions: (1) for each A, B ∈ B, there exists C ∈ B such that C ⊆ A ∩ B; (2) for all A ∈ B and a ∈ A, there exists B ∈ B such that aB ⊆ A; (3) B is subordinated to each A ∈ B. Conditions (1)–(3) imply that there exists a topology τ on H such that (H, τ ) is a semitopological group and the family B is a local base for H at eH . Since B is countable, (H, τ ) is first-countable. Let us show that the semitopological group (H, τ ) is T1 . Take y ∈ H \ {eH }, where eH is the identity of H. Choose x ∈ G such that p(x) = y. Since y = eH , we have that x ∈ / N = V ∈γ V . Therefore, there exists V ∈ γ such that x∈ / V = V N = p−1 p(V ). Hence y ∈ / p(V ). It follows that (H, τ ) is T1 . ∗ Finally, V0 = p(U0 ) is an open neighborhood of eH in H and p−1 (V0 ) = U0∗ N = U0∗ ⊆ U0 . 2
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In the case of projectively T0 first-countable semitopological groups, we don’t need a cardinal function. Theorem 2.13. A T0 semitopological group is projectively T0 first-countable if and only if it is ω-balanced and locally ω-good. 3. Open problems In [3], I. Guran showed that a Hausdorff topological group is topologically isomorphic to a subgroup of a product of second-countable topological groups if and only if it is ω-narrow. There exist internal characterizations of projectively (regular) Ti second-countable paratopological groups, for i = 0, 1, 2 (see [5] and [7]). In this direction, we propose the following problem. Problem 3.1. Give an internal characterization of projectively (regular) Ti second countable semitopological groups, for i = 0, 1, 2. By a result in [7], a regular paratopological group is projectively regular first-countable if and only if G is ω-balanced and Ir(G) ≤ ω. According to [2], every regular paratopological group is a Tychonoff space. So a Tychonoff paratopological group G is projectively Tychonoff first-countable if and only if G is ω-balanced and Ir(G) ≤ ω. This motivates the next problem. Problem 3.2. Give an internal characterization of projectively Tychonoff first-countable semitopological groups. Acknowledgements The author would like to give thanks to the family of Milcio Gutiérrez and Martha Lucia for their kindness and high quality human. References [1] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics, vol. I, Atlantis Press/World Scientific, Paris/Amsterdam, 2008. [2] T. Banakh, A. Ravsky, A characterization of completely regular spaces with applications to paratopological groups, http:// arxiv.org/abs/1410.1504. [3] I. Guran, Topological groups close to being Lindelöf, Sov. Math. Dokl. 23 (1981) 173–175; Russian original in: Dokl. Akad. Nauk SSSR 256 (1981) 1035–1037. [4] G.I. Katz, Isomorphic mapping of topological groups to direct product of groups satisfying the first axiom of countability, Usp. Mat. Nauk 8 (1953) 107–113 (in Russian). [5] I. Sánchez, Subgroups of products of paratopological groups, Topol. Appl. 163 (2014) 160–173. [6] M. Sanchis, M. Tkachenko, Totally Lindelöf and totally ω-narrow paratopological groups, Topol. Appl. 155 (2007) 322–334. [7] M.G. Tkachenko, Embedding paratopological groups into topological products, Topol. Appl. 156 (2009) 1298–1305.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Projectively first-countable semitopological groups Iván Sánchez Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Campus de Riu Sec s/n, Castelló, Spain
a r t i c l e
i n f o
Article history: Received 19 December 2014 Received in revised form 16 March 2016 Accepted 17 March 2016
a b s t r a c t We show that a regular semitopological group G is projectively regular firstcountable if and only if G is ω-balanced, locally ω-good and Ir(G) ≤ ω. Similarly, we give an internal characterization of projectively Ti first-countable semitopological groups, for i = 0, 1, 2. © 2016 Elsevier B.V. All rights reserved.
MSC: 54H11 54A25 54B10 54C25 54D10 Keywords: Semitopological groups Projectively first-countable Index of regularity Hausdorff number Symmetry number ω-balanced Locally ω-good
1. Introduction In 1953, G.I. Katz showed that a Hausdorff topological group G is topologically isomorphic to a subgroup of a product of first-countable (metrizable) topological groups if and only if G is ω-balanced (see [4]). As in [7], in this paper, a semitopological group G is regular if it is Hausdorff and for every open neighborhood U of the identity e in G, there exists an open neighborhood V of e such that V ⊆ U (usually this kind of spaces are called T3 ). Let P be a topological property. In [7], M. Tkachenko defines when a paratopological (topological) group G is projectively P. Similarly, we say that a semitopological group G is projectively P if for every neighborhood U of the identity in G, there exists a continuous homomorphism p : G → H onto a semitopological group
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.03.017 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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H with property P such that p−1 (V ) ⊆ U , for some neighborhood V of the neutral element in H. It is easy to see that a (regular, Tychonoff) Ti semitopological group is projectively (regular, Tychonoff) Ti first-countable if and only if it admits a homeomorphic embedding as a subgroup into a product of (regular, Tychonoff) Ti first-countable semitopological groups, for i = 0, 1, 2. M. Tkachenko extended Katz’s Theorem to the class of regular paratopological groups showing that a regular paratopological group G can be embedded as a subgroup of a product of regular first-countable paratopological groups if and only if G is ω-balanced and Ir(G) ≤ ω. In Theorem 2.9, we show that a regular semitopological group G is projectively regular first-countable if and only if G is ω-balanced, locally ω-good and Ir(G) ≤ ω. Similarly, in Theorems 2.11, 2.12 and 2.13, we give an internal characterization of projectively Ti first-countable semitopological groups, for i = 0, 1, 2. In [1], the reader can find the basic notions of semitopological (paratopological) groups. 2. Projectively first-countable semitopological groups Let G be a semitopological group with identity e. A subset V of G is called ω-good if there exists a countable family γ ⊂ N (e) such that for every x ∈ V , we can find W ∈ γ with xW ⊆ V . Denote by N ∗ (e) the family of ω-good sets of G which contain the identity. A semitopological group G with identity e is called locally ω-good if the family N ∗ (e) is a local base at e in G. According to the following result, every paratopological group is locally ω-good. Lemma 2.1. ([6, Lemma 3.10]) Every paratopological group G has a local base at the neutral element consisting of ω-good sets. The previous result cannot be extended to the class of T1 semitopological groups. Example 2.2. There exists a T1 compact semitopological group which is not locally ω-good. Proof. Consider the additive group R of real numbers. Let τ be the co-finite topology on R, i.e., τ = {U ⊆ R : |R \ U | < ω} ∪ {∅}. Clearly, G = (R, τ ) is T1 compact semitopological group. Let us show that G is not locally ω-good. Suppose the contrary that G is locally ω-good. Put U = G \ {1}. Then we can find V ∈ N ∗ (e) such that V ⊆ U . Clearly V is uncountable. Since V ∈ N ∗ (e), there exists a countable family γ = {Un ∈ N (e) : n ∈ ω} such that for every x ∈ V , we can choose n ∈ ω with xUn ⊆ V . We can assume that for each n ∈ ω, Un = G \ Fn , where Fn is a finite subset of G. Put F = n∈ω Fn . By the definition of τ , we have that V \ F = G \ A, where A is a countable subset of G. Hence there exists a ∈ G such that a, 1 − a ∈ V \ F . It follows that 1 = a + (1 − a) ∈ a + Un for every n ∈ ω. This contradiction shows that G is not locally ω-good. 2 In the following two propositions, we show that the property “locally ω-good” is closed with respect to subgroups and topological products. Proposition 2.3. Let {Gi : i ∈ I} be a family of locally ω-good semitopological groups. Then the product G = i∈I Gi is locally ω-good. Proof. Let U be an open neighborhood of the identity e in G. We can assume that U is a canonical open set, with U = i∈I Ui and Ui = Gi for i ∈ {i1 , . . . , in }. Let ei be the identity in Gi , for every i ∈ I. Since every Gi is locally ω-good, for each 1 ≤ k ≤ n, we can find Vik ∈ N ∗ (eik ) such that Vik ⊆ Uik . Put V = i∈I Wi , where Wi = Gi if i ∈ / {i1 , . . . , in } and Wik = Vik for each 1 ≤ k ≤ n. It is easy to see that V ∈ N ∗ (e) and V ⊆ U . This finishes the proof. 2
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Proposition 2.4. If H is a subgroup of a locally ω-good semitopological group G, then H is locally ω-good. Proof. Let U be an open neighborhood of the identity e in G. By hypothesis, there exists V ∈ N ∗ (e) such that V ⊆ U . Since V ∈ N ∗ (e), we can find a countable family γ ⊆ N (e) such that for every x ∈ V , there exists W ∈ γ satisfying xW ⊆ V . Put γ = {H ∩ W : W ∈ γ}. Clearly, γ is countable. Let us show that V ∩ H is an ω-good set of H. Indeed, fix h ∈ H ∩ V . Take W ∈ γ satisfying hW ⊆ V . We have h(H ∩ W ) = H ∩ hW ⊆ H ∩ V. Since H ∩ V is an ω-good set of H and H ∩ V ⊆ H ∩ U , H is locally ω-good.
2
The next result is evident. Proposition 2.5. Every first-countable semitopological group is locally ω-good. Corollary 2.6. Every subgroup of an arbitrary product of first-countable semitopological groups is locally ω-good. Proof. Apply Propositions 2.3–2.5.
2
Given a semitopological group G with identity e, we say that a family γ ⊆ N (e) is subordinated to a neighborhood U of the identity in G if for each x ∈ G there exists V ∈ γ such that xV x−1 ⊆ U . The semitopological group G is ω-balanced if for every U ∈ N (e), there exists a countable family γ subordinated to U . The following lemma plays an important role in this paper. Lemma 2.7. Let G be a semitopological group with identity e. Suppose that a family γ ⊂ N (e) satisfies the following conditions: (a) for every U ∈ γ and x ∈ U , there exists V ∈ γ such that xV ⊂ U ; (b) γ is subordinated to U , for each U ∈ γ. Then the set N = U ∈ γ.
{U ∩ U −1 : U ∈ γ} is an invariant subgroup of G. Further, U N = N U = U for each
Proof. It is clear that N = N −1 . Let us show that N is a subgroup of G. Take a, b ∈ N and U ∈ γ. It follows that a, b ∈ U ∩U −1 . By (a), there exists V ∈ γ such that aV ⊆ U . Hence ab ∈ aN ⊆ aV ⊆ U . Since b−1 ∈ U , we can find W ∈ γ satisfying b−1 W ⊆ U , equivalently, W −1 b ⊆ U −1 . We have that ab ∈ N b ⊆ W −1 b ⊆ U −1 . It follows that ab ∈ U ∩ U −1 for each U ∈ γ. Therefore, ab ∈ N . We have thus proved that N is a subgroup of G. Now, let us show that N is invariant in G. Take x ∈ G and U ∈ γ. By (b), we can find V ∈ γ such that xV x−1 ⊆ U . Hence xN x−1 ⊆ xV x−1 ⊆ U and xN x−1 ∈ xV −1 x−1 ⊆ U −1 . This implies that xN x−1 ∈ U ∩ U −1 for every U ∈ γ. This shows that N is invariant in G. For the last part, pick up U ∈ γ. Take a ∈ U . By (a), we can find V ∈ γ such that aV ⊆ U . It follows that aN ⊆ aV ⊆ U . Therefore, U N = U for each U ∈ γ. Since N is invariant in G, uN = N u for each u ∈ U ∈ γ. It follows that N U = U N = U for every U ∈ γ. 2 As in [7], in this paper, a semitopological group G is regular if it is Hausdorff and for every open neighborhood U of the identity e in G, there exists an open neighborhood V of e such that V ⊆ U
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(usually this kind of spaces are called T3 ). The following cardinal function permits us to give an internal characterization of projectively regular first-countable semitopological groups. Definition 2.8. ([7]) Denote by N (e) the family of neighborhoods of the identity e in a semitopological group G. We recall that the index of regularity Ir(G) of a regular semitopological group G is the minimum cardinal number κ such that for every neighborhood U of the identity e in G, we can find V ∈ N (e) and a family γ ⊆ N (e) such that W ∈γ V W −1 ⊆ U and |γ| ≤ κ. Theorem 2.9. Let G be a regular semitopological group. Then G is projectively regular first-countable if and only if G is ω-balanced, locally ω-good and Ir(G) ≤ ω. Proof. Suppose that G is projectively regular first-countable, then G is topologically isomorphic to a sub group of Π = i∈I Hi , where Hi is a regular first-countable semitopological group for each i ∈ I. By [7, Corollary 3.4], G is ω-balanced and Ir(G) ≤ ω. By Corollary 2.6, G is locally ω-good. Now, suppose that G is an ω-balanced, locally ω-good and Ir(G) ≤ ω. Let e be the identity of G. Take an arbitrary U0 ∈ N (e). We shall define a continuous homomorphism p : G → H onto a regular first-countable semitopological group H such that p−1 (V0 ) ⊆ U0 for some V0 ∈ N (eH ). We will construct by induction a sequence {γn : n ∈ ω} satisfying the following conditions for each n ∈ ω: (i) (ii) (iii) (iv) (v) (vi)
γn ⊆ N ∗ (e) and |γn | ≤ ω; γn ⊆ γn+1 ; γn is closed under finite intersections; the family γn+1 is subordinated to U , for each U ∈ γn ; for every U ∈ γn and x ∈ U , there exists V ∈ γn+1 such that xV ⊆ U ; for each U ∈ γn , there exists V ∈ γn+1 such that W ∈γn+1 V W −1 ⊆ U .
Since G is locally ω-good, there exists U0∗ ∈ N ∗ (e) such that U0∗ ⊆ U0 . Put γ0 = {U0∗ }. Suppose that for some n ∈ ω, we have defined families γ0 , . . . , γn satisfying (i)–(vi). Since γn is countable and G is ω-balanced, we can find a countable family λn,1 ⊆ N ∗ (e) subordinated to every U ∈ γn . As γn ⊆ N ∗ (e), there exists a countable family λn,2 ⊆ N ∗ (e) such that for each U ∈ γn and x ∈ U , there exists V ∈ λn,2 satisfying xV ⊆ U . Since Ir(G) ≤ ω, we can find a countable family λn,3 ⊆ N ∗ (e) such that for every U ∈ γn , there exists V ∈ λn,3 satisfying W ∈λn,3 V W −1 ⊆ U . Let γn+1 be the minimal family containing 3 γn ∪ i=1 λn,i and closed under finite intersections. Clearly, γn+1 is countable and satisfies (i)–(vi). This finishes our construction. Note that γ = n∈ω γn is countable and satisfies conditions (a)–(b) of Lemma 2.7. Hence, the set N = {V ∩ V −1 : V ∈ γ} is an invariant subgroup of G. It follows from the construction of γ that −1 ⊆ U , for every U ∈ γ, so N = V ∈γ V V −1 . Consider the algebraic group H = G/N . Let V ∈γ V V p : G → H be the canonical homomorphism. Put B = {p(V ) : V ∈ γ}. We claim that the family B satisfies the following properties: (1) for each A, B ∈ B, there exists C ∈ B such that C ⊆ A ∩ B; (2) for all A ∈ B and a ∈ A, there exists B ∈ B such that aB ⊆ A; (3) B is subordinated to each A ∈ B. Indeed, since γ is closed under finite intersections, condition (1) holds. Conditions (2) and (3) follow from (iv)–(v). Conditions (1)–(3) imply that there exists a topology τ on H such that (H, τ ) is a semitopological group and the family B is a local base for H at eH . Since B is countable, (H, τ ) is first-countable.
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Let us show that the semitopological group (H, τ ) is regular. First, we prove that (H, τ ) is Hausdorff. Take y ∈ H \ {eH }, where eH is the identity of H. Choose x ∈ G such that p(x) = y. Since y = eH , we have that x ∈ / N = V ∈γ V V −1 . Therefore, there exists V ∈ γ such that x ∈ / V V −1 , equivalently, xV ∩ V = ∅. It follows from Lemma 2.7 that xV N ∩ V N = ∅, so yp(V ) ∩ p(V ) = ∅. Now, choose U ∈ γ. Then U ∈ γn for some n ∈ ω. By (vi), we can find V ∈ γn+1 such that −1 ⊆ U . Let us show that p(V ) ⊆ p(U ). Take y ∈ H \ p(U ) and choose x ∈ G such that W ∈γn+1 V W p(x) = y. Clearly x ∈ / U . Therefore, there exists W ∈ γn+1 such that x ∈ / V W −1 , equivalently, xW ∩ V = ∅. It follows from Lemma 2.7 that xW N ∩ V N = ∅, so yp(W ) ∩ p(V ) = ∅. We have thus proved that y ∈ / p(V ). We conclude that (H, τ ) is regular. Finally, V0 = p(U0∗ ) is an open neighborhood of eH in H. It follows from Lemma 2.7 that p−1 (V0 ) = ∗ U0 N = U0∗ ⊆ U0 . 2 In Theorem 2.11, we generalize Katz’s theorem to the class of Hausdorff semitopological groups. First, we need the next cardinal function. Definition 2.10. ([7]) For a Hausdorff semitopological group G with identity e, the Hausdorff number of G, denoted by Hs(G), is the minimum cardinal number κ such that for every neighborhood U of e in G, there exists a family γ of neighborhoods of e such that V ∈γ V V −1 ⊆ U and |γ| ≤ κ. Theorem 2.11. A Hausdorff semitopological group is projectively Hausdorff first-countable if and only if it is ω-balanced, locally ω-good and has countable Hausdorff number. Proof. Let G be a Hausdorff semitopological group. Suppose that G is projectively Hausdorff first-countable, then G is topologically isomorphic to a subgroup of Π = i∈I Hi , where Hi is a Hausdorff first-countable semitopological group for each i ∈ I. Clearly, G is ω-balanced and locally ω-good. By Propositions 2.1–2.3 in [7], we have that Hs(G) ≤ ω. Now, suppose that G is an ω-balanced, locally ω-good and Hs(G) ≤ ω. Let e be the identity of G. Take an arbitrary U0 ∈ N (e). We shall define a continuous homomorphism p : G → H onto a Hausdorff first-countable semitopological group H such that p−1 (V0 ) ⊆ U0 for some V0 ∈ N (eH ). We will construct by induction a sequence {γn : n ∈ ω} satisfying the following conditions for each n ∈ ω: (i) (ii) (iii) (iv) (v) (vi)
γn ⊆ N ∗ (e) and |γn | ≤ ω; γn ⊆ γn+1 ; γn is closed under finite intersections; the family γn+1 is subordinated to U , for each U ∈ γn ; for every U ∈ γn and x ∈ U , there exists V ∈ γn+1 such that xV ⊆ U ; −1 ⊆ U , for each U ∈ γk . V ∈γn+1 V V
Since G is locally ω-good, we can find U0∗ ∈ N ∗ (e) such that U0∗ ⊆ U0 . Put γ0 = {U0∗ }. Suppose that for some n ∈ ω, we have defined families γ0 , . . . , γn satisfying (i)–(vi). Since γn is countable and G is ω-balanced, we can find a countable family λn,1 ⊆ N ∗ (e) subordinated to every U ∈ γn . As γn ⊆ N ∗ (e), there exists a countable family λn,2 ⊆ N ∗ (e) such that for each U ∈ γn and x ∈ U , we can choose V ∈ λn,2 satisfying xV ⊆ U . Since Hs(G) ≤ ω, we can find a countable family λn,3 ⊆ N ∗ (e) such that V ∈λn,3 V V −1 ⊆ U for 3 every U ∈ γn . Let γn+1 be the minimal family containing γn ∪ i=1 λn,i and closed under finite intersections. Clearly, γn+1 is countable and satisfies (i)–(vi). This finishes our construction. Note that γ = n∈ω γn is countable and satisfies conditions (a)–(b) of Lemma 2.7. Hence, the set N = {V ∩ V −1 : V ∈ γ} is an invariant subgroup of G. It follows from the construction of γ that V ∈γ V V −1 ⊆ U , for every U ∈ γ, so N = V ∈γ V V −1 . Consider the algebraic group H = G/N . Let p : G → H be the
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canonical homomorphism. Put B = {p(V ) : V ∈ γ}. Arguing as in the proof of Theorem 2.9, we conclude that there exists a topology τ on H such that (H, τ ) is a first-countable semitopological group and the family B is a local base for H at eH . Let us show that the semitopological group (H, τ ) is Hausdorff. Take y ∈ H \ {eH }, where eH is the identity of H. Choose x ∈ G such that p(x) = y. Since y = eH , we have that x ∈ / N = V ∈γ V V −1 . Therefore, there exists V ∈ γ such that x ∈ / V V −1 , equivalently, xV ∩ V = ∅. It follows from Lemma 2.7 that xV N ∩ V N = ∅, so yp(V ) ∩ p(V ) = ∅. We have thus proved that (H, τ ) is Hausdorff. Finally, V0 = p(U0∗ ) is an open neighborhood of eH in H and p−1 (V0 ) = U0∗ N = U0∗ ⊆ U0 . 2 The symmetry number of a T1 semitopological group G, denoted by Sm(G), as the minimum cardinal number κ such that for every neighborhood U of e in G, there exists a family γ ⊆ N (e) such that −1 ⊆ U and |γ| ≤ κ (see [5]). V ∈γ V Theorem 2.12. A T1 semitopological group G is projectively T1 first-countable if and only if G is ω-balanced, locally ω-good and Sm(G) ≤ ω. Proof. Suppose that G is projectively T1 first-countable. Then G is ω-balanced G and locally ω-good. By propositions 2.1–2.3 in [5], we have that Sm(G) ≤ ω. Now, suppose that G is ω-balanced, locally ω-good and Sm(G) ≤ ω. Let e be the identity of G. Take an arbitrary U0 ∈ N (e). We shall define a continuous homomorphism p : G → H onto a T1 first-countable semitopological group H such that p−1 (V0 ) ⊆ U0 for some V0 ∈ N (eH ). As in the proofs of Theorems 2.9 and 2.11, we can construct a sequence {γn : n ∈ ω} satisfying the following conditions for each n ∈ ω: (i) (ii) (iii) (iv) (v) (vi) (vii)
γ0 = {U0∗ }; γn ⊆ N ∗ (e) and |γn | ≤ ω; γn ⊆ γn+1 ; γn is closed under finite intersections; the family γn+1 is subordinated to U , for each U ∈ γn ; for every U ∈ γn and x ∈ U , there exists V ∈ γn+1 such that xV ⊆ U ; −1 ⊆ U , for every U ∈ γn . V ∈γn+1 V
The family γ = n∈ω γn is countable and satisfies conditions (a)–(b) of Lemma 2.7. This implies that N = {V ∩ V −1 : V ∈ γ} is an invariant subgroup of G. It follows from the construction of γ that N = V ∈γ V = V ∈γ V −1 . Consider the algebraic group H = G/N . Let p : G → H be the canonical homomorphism. Put B = {p(V ) : V ∈ γ}. Arguing as in the proof of Theorem 2.9, we conclude that the family B satisfies the following conditions: (1) for each A, B ∈ B, there exists C ∈ B such that C ⊆ A ∩ B; (2) for all A ∈ B and a ∈ A, there exists B ∈ B such that aB ⊆ A; (3) B is subordinated to each A ∈ B. Conditions (1)–(3) imply that there exists a topology τ on H such that (H, τ ) is a semitopological group and the family B is a local base for H at eH . Since B is countable, (H, τ ) is first-countable. Let us show that the semitopological group (H, τ ) is T1 . Take y ∈ H \ {eH }, where eH is the identity of H. Choose x ∈ G such that p(x) = y. Since y = eH , we have that x ∈ / N = V ∈γ V . Therefore, there exists V ∈ γ such that x∈ / V = V N = p−1 p(V ). Hence y ∈ / p(V ). It follows that (H, τ ) is T1 . ∗ Finally, V0 = p(U0 ) is an open neighborhood of eH in H and p−1 (V0 ) = U0∗ N = U0∗ ⊆ U0 . 2
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In the case of projectively T0 first-countable semitopological groups, we don’t need a cardinal function. Theorem 2.13. A T0 semitopological group is projectively T0 first-countable if and only if it is ω-balanced and locally ω-good. 3. Open problems In [3], I. Guran showed that a Hausdorff topological group is topologically isomorphic to a subgroup of a product of second-countable topological groups if and only if it is ω-narrow. There exist internal characterizations of projectively (regular) Ti second-countable paratopological groups, for i = 0, 1, 2 (see [5] and [7]). In this direction, we propose the following problem. Problem 3.1. Give an internal characterization of projectively (regular) Ti second countable semitopological groups, for i = 0, 1, 2. By a result in [7], a regular paratopological group is projectively regular first-countable if and only if G is ω-balanced and Ir(G) ≤ ω. According to [2], every regular paratopological group is a Tychonoff space. So a Tychonoff paratopological group G is projectively Tychonoff first-countable if and only if G is ω-balanced and Ir(G) ≤ ω. This motivates the next problem. Problem 3.2. Give an internal characterization of projectively Tychonoff first-countable semitopological groups. Acknowledgements The author would like to give thanks to the family of Milcio Gutiérrez and Martha Lucia for their kindness and high quality human. References [1] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics, vol. I, Atlantis Press/World Scientific, Paris/Amsterdam, 2008. [2] T. Banakh, A. Ravsky, A characterization of completely regular spaces with applications to paratopological groups, http:// arxiv.org/abs/1410.1504. [3] I. Guran, Topological groups close to being Lindelöf, Sov. Math. Dokl. 23 (1981) 173–175; Russian original in: Dokl. Akad. Nauk SSSR 256 (1981) 1035–1037. [4] G.I. Katz, Isomorphic mapping of topological groups to direct product of groups satisfying the first axiom of countability, Usp. Mat. Nauk 8 (1953) 107–113 (in Russian). [5] I. Sánchez, Subgroups of products of paratopological groups, Topol. Appl. 163 (2014) 160–173. [6] M. Sanchis, M. Tkachenko, Totally Lindelöf and totally ω-narrow paratopological groups, Topol. Appl. 155 (2007) 322–334. [7] M.G. Tkachenko, Embedding paratopological groups into topological products, Topol. Appl. 156 (2009) 1298–1305.