On precompact sets in spaces Cc(X)

Abstract

We show that the fact that X has a compact resolution swallowing the compact sets characterizes those C c .X/ spaces which have the so-called G-base. So, if X has a compact resolution which swallows all compact sets, then C c .X / belongs to the class G of Cascales and Orihuela (a large class of locally convex spaces which includes the (LM) and (DF)-spaces) for which all precompact sets are metrizable and, conversely, if C c .X/ belongs to the class G and X satisfies an additional mild condition, then X has a compact resolution which swallows all compact sets. This fully applicable result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space C c .X / in terms of topological properties of X and leads to a nice theorem of Cascales and Orihuela stating that for X containing a dense subspace with a compact resolution, every compact set in C c .X / is metrizable.In what follows, unless otherwise stated, X will be a Hausdorff completely regular space and C p .X/ and C c .X / will denote the space C.X/ of all real-valued continuous functions defined on X provided with the pointwise convergence topology and with the compact-open topology, respectively.Let us recall that a family A D ¹A˛W˛2 N N º of subsets of a set X is called a resolution of X if S ¹A˛W˛2 N N º D X and A˛Â Aˇfor˛Ä( coordinatewise), see [9, Chapter 3]. A resolution A of a topological space X is called compact if it consists of compact sets.A locally convex space (lcs) E is said to have a G-base (or a G-basis) (see [9, Chapter 1]) if there exists a basis ¹U˛W˛2 N N º of (absolutely convex) neighborhoods of the origin in E, such that UˇÂ U˛whenever˛Äˇ.An lcs E is said to belong to the class G if its topological dual E 0 has a resolution ¹A˛W˛2 N N º such that for every˛2 N N each sequence in A˛is

On topological properties of Fréchet locally convex spaces with the weak topology

“…Indeed, consider the locally compact space X = [0, ω 1 ). Under the assumption ω 1 = b, the space X has a compact resolution swallowing compact sets (see [17]). Every compact set in X, being countable, is metrizable.…”

“…Under (CH) the space X has a compact resolution swallowing compact sets by [39,Theorem 3.6]. Hence C c (X) is in class G by [17]. The space C c (X) is not in class G if we assume (M A + ¬CH), since by mentioned [39,Theorem 3.6] the space X even does not have a compact resolution, so by the same reason as above (use again [17]) the space C c (X) is not in class G.…”

Bounded sets structure of CpX and quasi‐(DF)‐spaces

“…Conversely, if X is σ‐compact, then has a fundamental compact resolution by Corollary 2.10 of . Once again applying Theorem 2 of , we obtain that E has a ‐base. To prove the last assertion we note that any metrizable σ‐compact space is an ℵ 0 ‐space, and hence is Lindelöf by .…”

“…Thus swallows the compact sets of . (ii)⇒(iii) follows from Theorem 2 of , and (iii) implies (i) since is a subspace of by Theorem 1.2 of . The last assertion follows from the fact that X is a subspace of and Cascales–Orihuela's theorem [, Theorem 11] (which states that every compact subset of an lcs with a ‐base is metrizable).…”

“…Therefore, by Proposition , E has a fundamental bounded resolution if and only if the space has a ‐base. But has a ‐base if and only if X has a fundamental compact resolution by .…”