On uniformly dense subspaces of function spaces
by
Vladimir V. Tkachuk
Universidad Autonoma Metropolitana de Mexico

It is a widespread method to obtain information about a topological space considering its dense subspaces. It has numerous applications in function spaces C_p(X) which have very rich algebraic and topological structure; as a result, quite a few properties hold in C_p(X) if discovered in a dense subspace of C_p(X). For example, if C_p(X) has a dense metrizable subspace then it is metrizable; if X is compact and C_p(X) has a dense Lindelöf \Sigma-subspace then C_p(X) is Lindelöf \Sigma. However, there are still many properties P which are not necessarily present in C_p(X) if some dense D subset C_p(X) has P. A good example is a discrete space X for which C_p(X)=R^X has a dense \sigma-compact Fréchet-Urysohn subspace while tightness of C_p(X) can be arbitrarily big. The same example shows that no compactness properties of X are implied by existence of a \sigma-compact (or even countable) dense subspace of C_p(X).

A set A subset C_p(X) is uniformly dense in C_p(X) if it is dense in the uniform topology on C(X), i.e., for any f in C_p(X) and any \epsilon > 0 there is g in A such that |g(x)-f(x)| < \epsilon for all x in X. This concept arises naturally if uniform and/or compact-open topologies are considered on function spaces; another important context is the area of applications of Stone-Weierstrass theorem. If we consider a uniformly dense subspace of C_p(X) then we have a much better approximation of C_p(X) than by a dense subspace. This gives hope that many properties of uniformly dense subspaces of C_p(X) imply themselves in C_p(X). We prove that this is indeed the case; another motivation for considering uniformly dense subspaces of C_p(X) is that they are quite helpful when countable decompositions of spaces C_p(X) are studied; one of the applications is the proof that every \sigma-metrizable C_p(X) is metrizable.

This work is an attempt of a systematic study of the relationship between C_p(X) and its uniformly dense subspaces. Our results show that a lot of new information can be obtained in this way about the spaces C_p(X). We prove that if C_p(X) has a uniformly dense \sigma-pseudocompact subspace then X is pseudocompact. If there is a \sigma-countably compact uniformly dense subspace of C_p(X) then X is compact and C_p(X) has a uniformly dense \sigma-compact subspace. There is a countable uniformly dense subspace of C_p(X) if and only if X is compact and metrizable. It turns out that if C_p(X) has a uniformly dense Lindelöf \Sigma-subspace then C_p(X) is Lindelöf \Sigma. The same is true for any given tightness, network weight and the Fréchet-Urysohn property. In fact, we show that if C_p(X) has a uniformly dense k-space then it is Fréchet-Urysohn; this strengthens a well-known theorem of Gerlitz-Nagy and Pytkeev. We also give examples of properties which are not implied in C_p(X) by their presence in uniformly dense subspaces of C_p(X). The most difficult one is an example of a compact non-separable space X for which C_p(X) has a uniformly dense subspace of countable pseudocharacter.

Date received: May 16, 2002