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Cardinal Functions, Part I
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Cardinal functions, Part I
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Cardinal Functions, Part I
A function ϕ which assigns a cardinal ϕ(X) to each topological space X is called a cardinal function or a cardinal invariant if it is a topological invariant, i.e., if we have ϕ(X) = ϕ(Y ) whenever X and Y are homeomorphic. Here we assume that the values of cardinal functions are always infinite cardinals. The simplicity of infinite cardinal arithmetic (i.e., κ + λ = κ · λ = λ whenever κ and λ are infinite cardinal numbers with κ λ) helps us to simplify our statements. We adopt the following set-theoretic notation: cardinal numbers are initial ordinals, i.e., κ is a cardinal if and only if it is the smallest ordinal of the cardinality |κ|. κ, λ, . . . always denote cardinal numbers. ω and ω1 are used to denote the first infinite ordinal (cardinal) and the first uncountable ordinal (cardinal) respectively. So, for each cardinal κ, we have κ + ω = κ if κ is infinite, and κ + ω = ω if κ is finite. To avoid typesetting problems, 2κ is often denoted by exp κ. Thus exp ω = 2ω is the cardinality c of the continuum, in other words, the cardinality of the set of real numbers. Let (X, τ ) be a topological space. Obviously, the cardinality |X| of X is the simplest cardinal function. Recall that a subfamily B of τ is a base if every member of τ is the union of a subfamily of B. The most important cardinal function is the weight w(X) of X, which is defined by w(X) = min{|B|: B is a base for X} + ω. Here ω is added to make the value infinite. A space X satisfies the second axiom of countability, or is second-countable if w(X) ω, in other words, if it has a countable base. In the early stage of general topology, first, Euclidean spaces and its subspaces were studied. In the following stage, separable metrizable spaces played an important role. Indeed, for example, Kuratowski mainly concentrated on separable metrizable spaces in his book [4]. Kinds of countability played an important role in the classical theory. For a regular T1 -space X, the following are equivalent: (a) X is separable metrizable, (b) w(X) = ω (i.e., X is second-countable), (c) X can be embedded in the product I ω of countably many copies of the unit interval I = [0, 1]. More generally, for a completely regular T1 -space X and an infinite cardinal κ, the following (b ) and (c ) are equivalent: (b ) w(X) = κ, (c ) X can be embedded in the product I κ of κ many copies of the unit interval I = [0, 1], which is called Tychonoff cube of weight κ. Hence weight is a cardinal function which measures, in some sense, the size of a typical space in which the space can be embedded. In Cardinal functions Part I, we give a brief introduction to simple and well-known cardinal functions, which are obtained as generalizations of classical countability axioms
(for more information see [3], [KV, Chapter 1], [E, Problems, Cardinal functions I, II, III, IV]). The inequalities between the cardinal functions defined here are summarized in [KV, Chapter 1, §3, Figure 1]. Now we turn our attention to other cardinal invariants. For a metrizable space X, the separability of X can be characterized by some other cardinal functions. Let X be a topological space. The Lindelöf degree (Lindelöf number) L(X) (sometimes denoted by l(X)) of X is defined by L(X) = min{κ: every open cover of X has a subcover of cardinality κ} + ω. If L(X) = ω, i.e., every open cover has a countable subcover, we say that X is Lindelöf. Define the density d(X) of X by d(X) = min{|D|: D is a dense subset of X}. If d(X) = ω, we say that X is a separable space. Define the cellularity (Souslin number) c(X) of X by c(X) = min{κ: every family of pairwise disjoint nonempty open subsets of X has cardinality κ} + ω. If c(X) = ω, we say that X has the countable chain condition (Souslin property). Countable chain condition is abbreviated as ccc. The spread s(X) and the extent e(X) are defined as follows: s(X) = sup{|D|: D is a discrete subset of X} + ω, and e(X) = sup{|D|: D is a discrete closed subset of X} + ω. For a metrizable space X, we have w(X) = L(X) = d(X) = c(X) = s(X) = e(X). For a completely metrizable space X, we have either |X| = w(X) or |X| = w(X)ω (all these results can be found in [KV, Chapter 1, §8]). Those cardinal functions are not equal for general topological spaces. However there are some relations between them. For example, we have the following inequalities: L(X) w(X), c(X) d(X) w(X), and d(X) |X|. We give here some examples showing the difference between such cardinal functions. The Sorgenfrey line S is defined to be the real line R as a set. The base for S is a family consisting of all half open intervals of the form [p, q) with p, q ∈ R and p < q. S satisfies L(S) = ω < 2ω = w(S). Let D be an uncountable discrete space and X be the one-point compactification of D. Then L(X) = ω < |D| = c(X). Let Y = I exp exp ω
(or {0, 1}exp exp ω )
be the Tychonoff product of exp exp ω many copies of the unit interval I = [0, 1] (or the two point discrete space {0, 1}), then c(Y ) = L(Y ) = ω, d(Y ) = exp ω and w(Y ) = exp exp ω. The cardinal functions defined above are called global cardinal functions. Now we introduce local cardinal functions. Let X be a topological space and x ∈ X. Recall that
12 a family U of neighbourhoods of x is called a neighbourhood base of x in X if for every neighbourhood V of x, there is U ∈ U with U ⊂ V . The character χ(x, X) of x in X is defined by χ(x, X) = min{|U|: U is a neighbourhood base of x in X} + ω. Furthermore the character χ(X) of X is defined by χ(X) = sup{χ(x, X): x ∈ X}. In other words, χ(X) = min{κ: κ ω, any point x ∈ X has a neighbourhood base of cardinality κ}. A space X satisfies the first axiom of countability, or is first-countable if χ(X) ω, i.e., if every point of X has a countable neighbourhood base. The tightness t (x, X) of x in X is defined by t (x, X) = min{κ: κ ω, for any set A ⊂ X with x ∈ cl A, there is a subset B of A with |B| κ and x ∈ cl B}, and define the tightness t (X) of X by t (X) = sup{t (x, X): x ∈ X}. In other words, t (X) = min{κ: κ ω, for any point x ∈ X and any subset A of X with x ∈ cl A, there is a subset B of A of cardinality κ satisfying x ∈ cl B}. Every metrizable space satisfies the first axiom of countability, and every first-countable space has countable tightness. For compact spaces, some cardinal functions can be characterized by the existence of a family with some weaker condition. A family N of subsets of a space (X, τ ) is called a network if every member of τ is a union of a subfamily of N , i.e., if for any V ∈ τ and any x ∈ V , there is A ∈ N with x ∈ A ⊂ V . Hence, a family N of subsets of X is a base for X if and only if it is a network for X consisting of open sets. Define the network weight nw(X) of X by nw(X) = min{|N |: N is a network for X} + ω. It is interesting to see that w(X) = nw(X) for any compact space X. Since the family {{x}: x ∈ X} is a network for any space X, we have w(X) = nw(X) |X| for any compact space X. Note that the equality w(X) = nw(X) also holds for any metrizable space X. Define the pseudocharacter ψ(x, X) of x in X by ψ(x, X) = min{|U|: U is a family of neighbourhoods of x with {x} = U}+ω. The pseudocharacter of X is defined by ψ(X) = sup{ψ(x, X): x ∈ X}. Then, by the definition of compactness, it is easy to see that χ(X) = ψ(X) for any compact T2 -space X. Some cardinal functions are bounded by some powers containing other cardinal functions. For example, we have |X| exp w(X) for any T1 -space X. This is because each point can be decided by the family of all open sets containing the point. It can be shown that w(X) exp d(X) for any regular space X. Indeed, let B be a base and D a dense set of a regular space X. Then {int cl B: B ∈ B} is a base consisting of regular open sets, where a regular open set is an open set U with U = int cl U . Since each regular open set can be decided by its intersection with D, we have the inequality. Some global cardinal functions are bounded by some combinations of global functions and local functions. For example, we have |X| d(X)χ(X) for any T2 -space X. In 1969, Arkhangel’ski˘ı proved a highly nontrivial result that the cardinality of any compact first-countable T2 -space is 2ω , answering Alexandroff and Urysohn’s problem that had been unanswered for about thirty years. More generally,
Section A: Generalities it can be shown that |X| exp L(X) · ψ(X) · t (X) for any T2 -space X. His original proof is difficult to understand. We give here the idea of a simplified proof due to R. Pol. Let X be a Lindelöf first-countable T2 -space. We show that |X| 2ω . Let Ux be a countable neighbourhood base for each x ∈ X, and A = A: A is a subset of X with |A| 2ω . Note that cl A ∈ A for any A ∈ A, because each point x ∈ cl A is decided by a sequence in A converging to x and there are 2ω many sequences in A. Consider an operation Φ which assigns to each closed set A ∈ A, an element Φ(A) ∈ A with the following property: A ⊂ Φ(A); andfor any countable subfamily V of x∈A U x of X with A ⊂ V and X − V = ∅, we have Φ(A) − V = ∅ (the Lindelöf property is used to show the existence of such V whenever X − A = ∅). An operation Φ exists because there are only 2ω manychoices of such V and so we can pick a point xV from X − V for such V. Let Φ(A) be the union of A and the set of such xV ’s. Now take any point x0 ∈ X and define A0 = {x0 }. By using transfinite induction, for each α < ω1 , define Aα = cl(Φ(cl( A )). Then we can show that β β<α Aα ∈ A and X = α<ω1 Aα , which implies |X| 2ω . The argument above is called “a closing-off argument” or “a closure argument”. Here, the term “closure” does not mean the topological closure but closure under a family of operations. A general way of simplifying closing-off arguments is via “elementary submodels of the universe”. For example, to prove Arkhangel’ski˘ı’s inequality by using an elementary submodel, just take one, M say, of cardinality 2ω such that X ∈ M and A ∈ M for any countable A ⊂ M. Here we need not mention the operation Φ above. To complete the proof, what we should do is show that X ⊂ M (see [2]). Now let us consider the behavior of cardinal functions under topological operations. It is interesting for us to observe that each cardinal function behaves quite differently. (1) continuous images: Let ϕ(X) be a cardinal function and f : X → Y be a continuous map onto Y . In this case, we say that ϕ is preserved by the map f if ϕ(Y ) ϕ(X). It is easy to see that the Lindelöf number L, density d and cellularity c are preserved under any continuous map. If X is compact, then weight w is preserved by any continuous map f : X → Y onto Y . Indeed, since X and Y are compact, we have w(X) = nw(X) and w(Y ) = nw(Y ). It is easy to check that network weight is preserved by any continuous map. However, generally speaking, weight w is not preserved under continuous maps. For example, let X = R2 and let Y be the space obtained from X by collapsing the x-axis to one point denoted by ∞. Let f : X → Y be the quotient map. Then X is second-countable and f is a closed map, but Y is not second-countable, because Y is not first-countable at ∞. At early stage of general topology, spaces satisfying some
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axiom of countability were studied. However uncountability naturally appears in such a situation. An easy way to describe this phenomenon is to consider sequential fans. The sequential fan S(κ) with κ spines is defined by the space obtained from the topological sum T (κ) of κ many copies of the convergent sequence by identifying all the limit points to a point denoted by ∞. Let f : T (κ) → S(κ) be the quotient map. If κ = ω, then f is a closed map from a secondcountable (hence, a separable metrizable) space T (ω) onto a space S(ω). But S(ω) is not first-countable at ∞. The character χ(∞, S(ω)) is equal to the cardinal denoted by d, and it can be shown that we cannot prove d = 2ω or d < 2ω by using our usual mathematics (i.e., ZFC set theory) (see [KV, Chapter 3]). (2) subspaces: A cardinal function ϕ is a monotone cardinal function if ϕ(Y ) ϕ(X) for any subspace Y of X. Weight w and character χ are monotone. But Lindelöf number L, density d and cellularity c are not monotone. For example, let D be an uncountable discrete space and X a onepoint compactification of D. Then L(D) > L(X). Let Y be the Niemytzki Plane and Z the x-axis of the plane. Then Z is a closed discrete set of Y and d(Z) = c(Z) = 2ω > ω = d(Y ) = c(Y ). For each cardinal function ϕ which is not monotone, define a new cardinal function hϕ by hϕ = sup{ϕ(Y ): Y ⊂ X}. Note that hc(X) = he(X) = s(X) for any space X. hL(X) and hd(X) will be discussed in Cardinal functions Part II. (3) products: Let ϕ be a cardinal function and κ, λ cardinals. The property ϕ κ is said to be a λ-multiplicative property if we have ϕ( α<λ Xα ) κ whenever ϕ(Xα ) κ for each α < λ. Likewise ϕ κ is said to be a multiplicative property if we have ϕ( α<λ Xα ) κ whenever λ is any cardinal and ϕ(Xα ) κ for each α < λ. It is easy to see that w κ is κ multiplicative. While compactness is multiplicative, the Lindelöf property is not multiplicative. The Sorgenfrey line S is Lindelöf, but the square S × S contains a closed discrete subset {(p, −p): p ∈ S} of cardinality 2ω , which implies that S × S is not Lindelöf. Next we consider density. For example, separability is c (= 2ω ) multiplicative but not c+ multiplicative. More generally, d κ is 2κ multiplicative (see [E, 2.3.15]), but not (2κ )+ multiplicative because of the inequality w(X) exp d(X). Surprisingly, the multiplicativity of cellularity depends on your set theory. It is known that a product has the countable chain condition if and only if any finite subproduct has the countable chain condition. So the point is whether cellularity is finitely multiplicative or not. Under Martin’s Axiom (more precisely MA(ℵ1 )), any Tyhonoff product of spaces with the countable chain condition has the countable chain condition (see [Ku, Chapter 2, 2.24]). On the other hand, assume the negation of the Souslin hypothesis, i.e., there is a Souslin tree, which is an ω1 -tree with no uncountable antichains and no uncountable branches. For example, if we
13 assume ♦, then there is such a tree. The set theoretical axiom ♦ holds under the assumption V = L (i.e., every set is constructible). The negation of the Souslin hypothesis is equivalent to the existence of a Souslin line L, that is a nonseparable linearly ordered topological space with the countable chain condition. The square L×L of a Souslin line does not satisfies the countable chain condition (see [Ku, Chapter 2, 4.3]). For more results on chain conditions, see [6]. (4) compactifications: For any compactification Y of X, we have |Y | exp exp d(X) and w(Y ) exp d(X). For example, let D(κ) be the infinite discrete space of cardinalˇ ity κ, and βD(κ) be the Cech–Stone compactification of D(κ). Then |βD(κ)| = exp exp κ and w(βD(κ)) = exp κ (see [E, 3.5, 3.6]). (5) function spaces (see [5, Chapter 4], [1]): We mention here only some typical results. All spaces here are assumed to be completely regular T1 -spaces. Let Ck (X, Y ) (respectively Cp (X, Y )) be the space of continuous functions from X to Y with the compact-open topology (respectively the topology of pointwise convergence). Ck (X, R) (respectively Cp (X, R)) is sometimes denoted by Ck (X) (respectively Cp (X)), while Cu (X) is the space of continuous real functions on X with the topology of uniform convergence. Then |X| = χ(Cp (X)) = w(Cp (X)) for any space X with |X| ω, and nw Cp (X, Y ) nw Ck (X, Y ) w(X) · w(Y ). There are beautiful relations between tightness and Lindelöf number as follows: For any space X and any n ∈ N , we have t (Xn ) L(Cp (X)) for any n ∈ N . Furthermore t (Cp (X)) κ if and only if L(Xn ) κ for any n ∈ N . A kind of duality between hereditary density and hereditary Lindelöf degree holds: n sup hd Xn : n ∈ N = sup hL Cp (X) : n ∈ N , and n sup hL Xn : n ∈ N = sup hd Cp (X) : n ∈ N . We have already mentioned the relations between cardinal functions for metrizable spaces and for compact spaces. Now we consider the relations between cardinal functions for other classes of spaces. (1) LOTS: For any linearly ordered topological space (LOTS) X, we have χ(X) = ψ(X) c(X) d(X) w(X) = nw(X) |X| (see [E, 3.12.4]). (2) Topological groups (see [KV, Chapter 24, §3]): For a topological group G, χ(G) = ω holds if and only if G is metrizable. For a topological group G, we have w(G) = d(G) · χ(G). For every compact infinite topological group G with w(G) = κ, we have |G| = exp κ, d(G) = log κ, and c(G) = ω. (3) Dyadic spaces (see [E, 3.12.12], [KV, Chapter 24, §1]): A compact space X is called a dyadic space if it is a continuous image of the Cantor cube D κ for some infinite κ, where D = {0, 1} with the discrete topology. It is
14 known that compact topological groups are dyadic. For any dyadic space X, we have c(X) = ω and surprisingly, we have w(X) = χ(X) = t (X).
References [1] A.V. Arkhangel’skii, Topological Function Spaces, Kluwer Academic, Dordrecht (1992). [2] A. Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), 17– 72.
Section A: Generalities [3] I. Juhász, Cardinal Functions in Topology – Ten Years Later, Mathematical Centre Tracts, Math. Centrum, Amsterdam (1983). [4] K. Kuratowski, Topology, Vol. I, Academic Press, New York (1966). [5] R.A. McCoy and I. Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Math., Vol. 1315, Springer, Berlin (1988). [6] S. Todorˇcevi´c, Chain-condition methods in topology, Topology Appl. 101 (2000), 45–82.
Kenichi Tamano Yokohama, Japan
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Cardinal Functions, Part I
A function ϕ which assigns a cardinal ϕ(X) to each topological space X is called a cardinal function or a cardinal invariant if it is a topological invariant, i.e., if we have ϕ(X) = ϕ(Y ) whenever X and Y are homeomorphic. Here we assume that the values of cardinal functions are always infinite cardinals. The simplicity of infinite cardinal arithmetic (i.e., κ + λ = κ · λ = λ whenever κ and λ are infinite cardinal numbers with κ λ) helps us to simplify our statements. We adopt the following set-theoretic notation: cardinal numbers are initial ordinals, i.e., κ is a cardinal if and only if it is the smallest ordinal of the cardinality |κ|. κ, λ, . . . always denote cardinal numbers. ω and ω1 are used to denote the first infinite ordinal (cardinal) and the first uncountable ordinal (cardinal) respectively. So, for each cardinal κ, we have κ + ω = κ if κ is infinite, and κ + ω = ω if κ is finite. To avoid typesetting problems, 2κ is often denoted by exp κ. Thus exp ω = 2ω is the cardinality c of the continuum, in other words, the cardinality of the set of real numbers. Let (X, τ ) be a topological space. Obviously, the cardinality |X| of X is the simplest cardinal function. Recall that a subfamily B of τ is a base if every member of τ is the union of a subfamily of B. The most important cardinal function is the weight w(X) of X, which is defined by w(X) = min{|B|: B is a base for X} + ω. Here ω is added to make the value infinite. A space X satisfies the second axiom of countability, or is second-countable if w(X) ω, in other words, if it has a countable base. In the early stage of general topology, first, Euclidean spaces and its subspaces were studied. In the following stage, separable metrizable spaces played an important role. Indeed, for example, Kuratowski mainly concentrated on separable metrizable spaces in his book [4]. Kinds of countability played an important role in the classical theory. For a regular T1 -space X, the following are equivalent: (a) X is separable metrizable, (b) w(X) = ω (i.e., X is second-countable), (c) X can be embedded in the product I ω of countably many copies of the unit interval I = [0, 1]. More generally, for a completely regular T1 -space X and an infinite cardinal κ, the following (b ) and (c ) are equivalent: (b ) w(X) = κ, (c ) X can be embedded in the product I κ of κ many copies of the unit interval I = [0, 1], which is called Tychonoff cube of weight κ. Hence weight is a cardinal function which measures, in some sense, the size of a typical space in which the space can be embedded. In Cardinal functions Part I, we give a brief introduction to simple and well-known cardinal functions, which are obtained as generalizations of classical countability axioms
(for more information see [3], [KV, Chapter 1], [E, Problems, Cardinal functions I, II, III, IV]). The inequalities between the cardinal functions defined here are summarized in [KV, Chapter 1, §3, Figure 1]. Now we turn our attention to other cardinal invariants. For a metrizable space X, the separability of X can be characterized by some other cardinal functions. Let X be a topological space. The Lindelöf degree (Lindelöf number) L(X) (sometimes denoted by l(X)) of X is defined by L(X) = min{κ: every open cover of X has a subcover of cardinality κ} + ω. If L(X) = ω, i.e., every open cover has a countable subcover, we say that X is Lindelöf. Define the density d(X) of X by d(X) = min{|D|: D is a dense subset of X}. If d(X) = ω, we say that X is a separable space. Define the cellularity (Souslin number) c(X) of X by c(X) = min{κ: every family of pairwise disjoint nonempty open subsets of X has cardinality κ} + ω. If c(X) = ω, we say that X has the countable chain condition (Souslin property). Countable chain condition is abbreviated as ccc. The spread s(X) and the extent e(X) are defined as follows: s(X) = sup{|D|: D is a discrete subset of X} + ω, and e(X) = sup{|D|: D is a discrete closed subset of X} + ω. For a metrizable space X, we have w(X) = L(X) = d(X) = c(X) = s(X) = e(X). For a completely metrizable space X, we have either |X| = w(X) or |X| = w(X)ω (all these results can be found in [KV, Chapter 1, §8]). Those cardinal functions are not equal for general topological spaces. However there are some relations between them. For example, we have the following inequalities: L(X) w(X), c(X) d(X) w(X), and d(X) |X|. We give here some examples showing the difference between such cardinal functions. The Sorgenfrey line S is defined to be the real line R as a set. The base for S is a family consisting of all half open intervals of the form [p, q) with p, q ∈ R and p < q. S satisfies L(S) = ω < 2ω = w(S). Let D be an uncountable discrete space and X be the one-point compactification of D. Then L(X) = ω < |D| = c(X). Let Y = I exp exp ω
(or {0, 1}exp exp ω )
be the Tychonoff product of exp exp ω many copies of the unit interval I = [0, 1] (or the two point discrete space {0, 1}), then c(Y ) = L(Y ) = ω, d(Y ) = exp ω and w(Y ) = exp exp ω. The cardinal functions defined above are called global cardinal functions. Now we introduce local cardinal functions. Let X be a topological space and x ∈ X. Recall that
12 a family U of neighbourhoods of x is called a neighbourhood base of x in X if for every neighbourhood V of x, there is U ∈ U with U ⊂ V . The character χ(x, X) of x in X is defined by χ(x, X) = min{|U|: U is a neighbourhood base of x in X} + ω. Furthermore the character χ(X) of X is defined by χ(X) = sup{χ(x, X): x ∈ X}. In other words, χ(X) = min{κ: κ ω, any point x ∈ X has a neighbourhood base of cardinality κ}. A space X satisfies the first axiom of countability, or is first-countable if χ(X) ω, i.e., if every point of X has a countable neighbourhood base. The tightness t (x, X) of x in X is defined by t (x, X) = min{κ: κ ω, for any set A ⊂ X with x ∈ cl A, there is a subset B of A with |B| κ and x ∈ cl B}, and define the tightness t (X) of X by t (X) = sup{t (x, X): x ∈ X}. In other words, t (X) = min{κ: κ ω, for any point x ∈ X and any subset A of X with x ∈ cl A, there is a subset B of A of cardinality κ satisfying x ∈ cl B}. Every metrizable space satisfies the first axiom of countability, and every first-countable space has countable tightness. For compact spaces, some cardinal functions can be characterized by the existence of a family with some weaker condition. A family N of subsets of a space (X, τ ) is called a network if every member of τ is a union of a subfamily of N , i.e., if for any V ∈ τ and any x ∈ V , there is A ∈ N with x ∈ A ⊂ V . Hence, a family N of subsets of X is a base for X if and only if it is a network for X consisting of open sets. Define the network weight nw(X) of X by nw(X) = min{|N |: N is a network for X} + ω. It is interesting to see that w(X) = nw(X) for any compact space X. Since the family {{x}: x ∈ X} is a network for any space X, we have w(X) = nw(X) |X| for any compact space X. Note that the equality w(X) = nw(X) also holds for any metrizable space X. Define the pseudocharacter ψ(x, X) of x in X by ψ(x, X) = min{|U|: U is a family of neighbourhoods of x with {x} = U}+ω. The pseudocharacter of X is defined by ψ(X) = sup{ψ(x, X): x ∈ X}. Then, by the definition of compactness, it is easy to see that χ(X) = ψ(X) for any compact T2 -space X. Some cardinal functions are bounded by some powers containing other cardinal functions. For example, we have |X| exp w(X) for any T1 -space X. This is because each point can be decided by the family of all open sets containing the point. It can be shown that w(X) exp d(X) for any regular space X. Indeed, let B be a base and D a dense set of a regular space X. Then {int cl B: B ∈ B} is a base consisting of regular open sets, where a regular open set is an open set U with U = int cl U . Since each regular open set can be decided by its intersection with D, we have the inequality. Some global cardinal functions are bounded by some combinations of global functions and local functions. For example, we have |X| d(X)χ(X) for any T2 -space X. In 1969, Arkhangel’ski˘ı proved a highly nontrivial result that the cardinality of any compact first-countable T2 -space is 2ω , answering Alexandroff and Urysohn’s problem that had been unanswered for about thirty years. More generally,
Section A: Generalities it can be shown that |X| exp L(X) · ψ(X) · t (X) for any T2 -space X. His original proof is difficult to understand. We give here the idea of a simplified proof due to R. Pol. Let X be a Lindelöf first-countable T2 -space. We show that |X| 2ω . Let Ux be a countable neighbourhood base for each x ∈ X, and A = A: A is a subset of X with |A| 2ω . Note that cl A ∈ A for any A ∈ A, because each point x ∈ cl A is decided by a sequence in A converging to x and there are 2ω many sequences in A. Consider an operation Φ which assigns to each closed set A ∈ A, an element Φ(A) ∈ A with the following property: A ⊂ Φ(A); andfor any countable subfamily V of x∈A U x of X with A ⊂ V and X − V = ∅, we have Φ(A) − V = ∅ (the Lindelöf property is used to show the existence of such V whenever X − A = ∅). An operation Φ exists because there are only 2ω manychoices of such V and so we can pick a point xV from X − V for such V. Let Φ(A) be the union of A and the set of such xV ’s. Now take any point x0 ∈ X and define A0 = {x0 }. By using transfinite induction, for each α < ω1 , define Aα = cl(Φ(cl( A )). Then we can show that β β<α Aα ∈ A and X = α<ω1 Aα , which implies |X| 2ω . The argument above is called “a closing-off argument” or “a closure argument”. Here, the term “closure” does not mean the topological closure but closure under a family of operations. A general way of simplifying closing-off arguments is via “elementary submodels of the universe”. For example, to prove Arkhangel’ski˘ı’s inequality by using an elementary submodel, just take one, M say, of cardinality 2ω such that X ∈ M and A ∈ M for any countable A ⊂ M. Here we need not mention the operation Φ above. To complete the proof, what we should do is show that X ⊂ M (see [2]). Now let us consider the behavior of cardinal functions under topological operations. It is interesting for us to observe that each cardinal function behaves quite differently. (1) continuous images: Let ϕ(X) be a cardinal function and f : X → Y be a continuous map onto Y . In this case, we say that ϕ is preserved by the map f if ϕ(Y ) ϕ(X). It is easy to see that the Lindelöf number L, density d and cellularity c are preserved under any continuous map. If X is compact, then weight w is preserved by any continuous map f : X → Y onto Y . Indeed, since X and Y are compact, we have w(X) = nw(X) and w(Y ) = nw(Y ). It is easy to check that network weight is preserved by any continuous map. However, generally speaking, weight w is not preserved under continuous maps. For example, let X = R2 and let Y be the space obtained from X by collapsing the x-axis to one point denoted by ∞. Let f : X → Y be the quotient map. Then X is second-countable and f is a closed map, but Y is not second-countable, because Y is not first-countable at ∞. At early stage of general topology, spaces satisfying some
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Cardinal functions, Part I
axiom of countability were studied. However uncountability naturally appears in such a situation. An easy way to describe this phenomenon is to consider sequential fans. The sequential fan S(κ) with κ spines is defined by the space obtained from the topological sum T (κ) of κ many copies of the convergent sequence by identifying all the limit points to a point denoted by ∞. Let f : T (κ) → S(κ) be the quotient map. If κ = ω, then f is a closed map from a secondcountable (hence, a separable metrizable) space T (ω) onto a space S(ω). But S(ω) is not first-countable at ∞. The character χ(∞, S(ω)) is equal to the cardinal denoted by d, and it can be shown that we cannot prove d = 2ω or d < 2ω by using our usual mathematics (i.e., ZFC set theory) (see [KV, Chapter 3]). (2) subspaces: A cardinal function ϕ is a monotone cardinal function if ϕ(Y ) ϕ(X) for any subspace Y of X. Weight w and character χ are monotone. But Lindelöf number L, density d and cellularity c are not monotone. For example, let D be an uncountable discrete space and X a onepoint compactification of D. Then L(D) > L(X). Let Y be the Niemytzki Plane and Z the x-axis of the plane. Then Z is a closed discrete set of Y and d(Z) = c(Z) = 2ω > ω = d(Y ) = c(Y ). For each cardinal function ϕ which is not monotone, define a new cardinal function hϕ by hϕ = sup{ϕ(Y ): Y ⊂ X}. Note that hc(X) = he(X) = s(X) for any space X. hL(X) and hd(X) will be discussed in Cardinal functions Part II. (3) products: Let ϕ be a cardinal function and κ, λ cardinals. The property ϕ κ is said to be a λ-multiplicative property if we have ϕ( α<λ Xα ) κ whenever ϕ(Xα ) κ for each α < λ. Likewise ϕ κ is said to be a multiplicative property if we have ϕ( α<λ Xα ) κ whenever λ is any cardinal and ϕ(Xα ) κ for each α < λ. It is easy to see that w κ is κ multiplicative. While compactness is multiplicative, the Lindelöf property is not multiplicative. The Sorgenfrey line S is Lindelöf, but the square S × S contains a closed discrete subset {(p, −p): p ∈ S} of cardinality 2ω , which implies that S × S is not Lindelöf. Next we consider density. For example, separability is c (= 2ω ) multiplicative but not c+ multiplicative. More generally, d κ is 2κ multiplicative (see [E, 2.3.15]), but not (2κ )+ multiplicative because of the inequality w(X) exp d(X). Surprisingly, the multiplicativity of cellularity depends on your set theory. It is known that a product has the countable chain condition if and only if any finite subproduct has the countable chain condition. So the point is whether cellularity is finitely multiplicative or not. Under Martin’s Axiom (more precisely MA(ℵ1 )), any Tyhonoff product of spaces with the countable chain condition has the countable chain condition (see [Ku, Chapter 2, 2.24]). On the other hand, assume the negation of the Souslin hypothesis, i.e., there is a Souslin tree, which is an ω1 -tree with no uncountable antichains and no uncountable branches. For example, if we
13 assume ♦, then there is such a tree. The set theoretical axiom ♦ holds under the assumption V = L (i.e., every set is constructible). The negation of the Souslin hypothesis is equivalent to the existence of a Souslin line L, that is a nonseparable linearly ordered topological space with the countable chain condition. The square L×L of a Souslin line does not satisfies the countable chain condition (see [Ku, Chapter 2, 4.3]). For more results on chain conditions, see [6]. (4) compactifications: For any compactification Y of X, we have |Y | exp exp d(X) and w(Y ) exp d(X). For example, let D(κ) be the infinite discrete space of cardinalˇ ity κ, and βD(κ) be the Cech–Stone compactification of D(κ). Then |βD(κ)| = exp exp κ and w(βD(κ)) = exp κ (see [E, 3.5, 3.6]). (5) function spaces (see [5, Chapter 4], [1]): We mention here only some typical results. All spaces here are assumed to be completely regular T1 -spaces. Let Ck (X, Y ) (respectively Cp (X, Y )) be the space of continuous functions from X to Y with the compact-open topology (respectively the topology of pointwise convergence). Ck (X, R) (respectively Cp (X, R)) is sometimes denoted by Ck (X) (respectively Cp (X)), while Cu (X) is the space of continuous real functions on X with the topology of uniform convergence. Then |X| = χ(Cp (X)) = w(Cp (X)) for any space X with |X| ω, and nw Cp (X, Y ) nw Ck (X, Y ) w(X) · w(Y ). There are beautiful relations between tightness and Lindelöf number as follows: For any space X and any n ∈ N , we have t (Xn ) L(Cp (X)) for any n ∈ N . Furthermore t (Cp (X)) κ if and only if L(Xn ) κ for any n ∈ N . A kind of duality between hereditary density and hereditary Lindelöf degree holds: n sup hd Xn : n ∈ N = sup hL Cp (X) : n ∈ N , and n sup hL Xn : n ∈ N = sup hd Cp (X) : n ∈ N . We have already mentioned the relations between cardinal functions for metrizable spaces and for compact spaces. Now we consider the relations between cardinal functions for other classes of spaces. (1) LOTS: For any linearly ordered topological space (LOTS) X, we have χ(X) = ψ(X) c(X) d(X) w(X) = nw(X) |X| (see [E, 3.12.4]). (2) Topological groups (see [KV, Chapter 24, §3]): For a topological group G, χ(G) = ω holds if and only if G is metrizable. For a topological group G, we have w(G) = d(G) · χ(G). For every compact infinite topological group G with w(G) = κ, we have |G| = exp κ, d(G) = log κ, and c(G) = ω. (3) Dyadic spaces (see [E, 3.12.12], [KV, Chapter 24, §1]): A compact space X is called a dyadic space if it is a continuous image of the Cantor cube D κ for some infinite κ, where D = {0, 1} with the discrete topology. It is
14 known that compact topological groups are dyadic. For any dyadic space X, we have c(X) = ω and surprisingly, we have w(X) = χ(X) = t (X).
References [1] A.V. Arkhangel’skii, Topological Function Spaces, Kluwer Academic, Dordrecht (1992). [2] A. Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), 17– 72.
Section A: Generalities [3] I. Juhász, Cardinal Functions in Topology – Ten Years Later, Mathematical Centre Tracts, Math. Centrum, Amsterdam (1983). [4] K. Kuratowski, Topology, Vol. I, Academic Press, New York (1966). [5] R.A. McCoy and I. Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Math., Vol. 1315, Springer, Berlin (1988). [6] S. Todorˇcevi´c, Chain-condition methods in topology, Topology Appl. 101 (2000), 45–82.
Kenichi Tamano Yokohama, Japan