The Density Character of Function Spaces

Abstract

Abstract.For topologies between the pointwise topology and the compact-open topology, the density character of C{X) (and, for certain spaces Z, C(X, Z)) is described in terms of a cardinal invariant of X. The Hewitt-Pondiczery theorem on the density character of product spaces follows as a corollary.1. Description. Except in Corollary 2, all hypothesized spaces are assumed to be completely regular Hausdorff. The set of continuous functions from X to Z is denoted by C(X, Z) or, when Z=R, by C(X).The density character, ÔX, of a space X is the least cardinality of a dense subset of X and the weight, wX, of X is the least cardinality of an open basis of X. We define the weak weight, wwX, of X to be the least of the cardinals w Y for Y a continuous one-to-one image of X. Theorem 1. Let X be any infinite space and let C(X) have any topology between the pointwise topology and the compact-open topology. Then ÔC(X) = wwX.All proofs will be given in the next section. Our remaining results allow Theorem 1 to be applied to yield information about 6C(X, Z) for suitable paces Z.Lemma. Let C(X,Z) and C(X,Z*) both have either the topology of uniform convergence on members of some fixed cover of X or the set-open topology generated by such a cover. IfZ is a retract ofZ* then ÔC(X, Z)Ô C(X, Z*). Proposition.For any topologies between the pointwise topology and the compact-open topology:(i) For any space Z, ôZ^ôC(X, Z).(ii) IfZ contains a nondegenerate arc, then ôC(X)^ôC(X, Z). (iii) If for each finite subset F of X there exists a function f in C(X,Z) such that F and f(F) have the same cardinality, then ôC(X)^ôC(X, Z) • ÔC(Z).

On topological spaces and topological groups with certain local countable networks

“…It is known (see [36]) that E 0 is Fréchet-Urysohn. Assume that E has a fundamental sequence (B n ) n of bounded sets, so E 0 admits such a sequence, too.…”

Variations of selective separability and tightness in function spaces with set-open topologies

“…Note that f (λ) is the family of compact subsets of f (X) and it is closed under compact subsets f (X) of its elements. By Theorem (N.Noble, [17]), the space C c (f (X)) is separable space then C f (λ) (f (X)) is a separable space. It follows immediately that C λ (X) is a separable space.…”

Covering properties of Cp(X) and Ck(X)

“…Since d C p (X) = iw (X) = w (X) = ℵ 1 [58], there is F 0 ⊆ F such that |F 0 | ≤ ℵ 1 and C p (X) = F 0 . This covering can be rewritten as {F α : 0 ≤ α < ω 1 }, and if we define G α = F β : 0 ≤ β < α for every 0 ≤ α < ω 1 , clearly G = {G α : 0 ≤ α < ω 1 } is an increasing closure-preserving covering of C p (X) by separable subspaces which swallows the separable sets in C p (X).…”