Embeddings and frames of function spaces

Abstract

For every Tychonoff space X we denote by C p (X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspaceembeds in a σ -compact space of countable tightness then X is countable. This shows that it is natural to study when C p (X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space C p (X) has a Lindelöf frame of countable tightness then t( X) ω. We give some generalizations of this result for the case of frames as well as for embeddings of C p (X) in arbitrary spaces.© 2009 Elsevier Inc. All rights reserved. IntroductionThe space C p (X) seldom has a compactness-like property; for example, C p (X) is a countable union of its countably compact subspaces if and only if X is finite [14]. However, for any pseudocompact space X it is possible to find a σ -compact frame for C p (X), i.e., there exists a σ -compact set P such that C p (X) ⊂ P ⊂ R X . It is a result of Uspenskij that for any Lindelöf Σ -space X there exists a Lindelöf Σ -space P which is a frame for C p (X). These results show the importance of studying the spaces X for which nice frames for C p (X) exist. Arhangel'skii [2] and Okunev [13] were the first ones to study and systematize the relevant classes of spaces. In [13], Okunev gave characterizations, in terms of topology of X , for existence of σ -compact frames, Lindelöf Σ -frames and many other kinds of frames for C p (X).Any result about nice frames of C p (X) actually says something about embeddings of C p (X) and sometimes, a possibility to embed C p (X) in a space with nice properties implies strong restrictions on X . We show that if C p (X) embeds in a σ -compact space of countable tightness then X is countable. This illustrates the importance of studying frames of countable tightness for C p (X).A theorem of Asanov [5] states that if C p (X) is Lindelöf then tightness of X is countable. It would be natural to try to obtain the same conclusion for the spaces X with a Lindelöf frame of countable tightness for C p (X). We prove that if X is a compact space and C p (X) has a Lindelöf frame of countable tightness then t( X) ω; an example is constructed which shows that compactness of X cannot be omitted in this result.We also establish that if the space C p (X) can be ω-embedded in a σ -compact space Z (this means that every point of Z is in the closure of a countable subset of C p (X)) then X has a small diagonal. Therefore, if X is a Lindelöf Σ -space and C p (X) can be ω-embedded in a σ -compact space then, under the Continuum Hypothesis, the space X has a countable network.