Spaces determined by point-countable covers

Abstract

Recall that a collection 9 of subsets of X is point-countable if every x G X is in at most countably many ?Gf. Such collections have been studied from several points of view. First, in characterizing various kinds of j-images of metric spaces, second, to construct conditions which imply that compact spaces and some of their generalizations are metrizable, and finally, in the context of meta-Lindelόf spaces. This paper will make some contributions to all of these areas. Introduction. A space1 X is determined by a cover £P, or 9 determines X, 2 if U C X is open (closed) in X if and only if U Π P is relatively open (relatively closed) in P for every P E <•?. (For example, X is determined by <$ if <$ is locally finite or if {P°: P E <3>} covers X.) We will explore this notion in the context of point-countable covers.Within the class of λ -spaces, spaces determined by point-countable covers turn out to be related to spaces having a point-countable Λ -network. Recall that a cover ζP of X is a k-network for X if, whenever K C U with K compact and U open in X 9 then K C U φ C U for some finitê C^. Such collections have played a role in S 0 -spaces (i.e., regular spaces with a countable λ -network [MJ) and K-spaces (i.e., regular spaces with a σ-locally finite λ -network [0]).

Metrizability of sequential topological groups with point-countable 𝑘-networks

“…By [N] it is an α 4 -space and thus is a countably bi-k-space in the sense of [M2]. Now by [GMT,Corollary 3.6] G is first countable. Thus by the classical Birkhoff-Kakutani theorem G is metrizable.…”

Decompositions of spaces determined by compact subsets

“…But Y is singly bi-fc, hence is Fréchet by Lemma 1.2. Then, since Y is regular, Y is paracompact by [8,Corollary 8.9]. As seen above, then, Y is tr-metric and yi is metric.…”

“…Let C be a cover of a space Z. Then Z is determined by C [8], or Z has the weak topology with respect to C, if F c Z is closed in Z if and only if F n C is relatively closed in C for every G G C. Here we can replace "closed" by "open". A space is a k-space if it is determined by the cover of all compact subsets.…”

A Characterization of Closed Images of Metric Spaces

“…It is now easy to see that @¿ = { P\D": P g 3dn} u {{x}: x g D"} is point-finite, hereditarily closurepreserving and that U^n^,, ' satisfies the k-network property. This answers a question of [5].…”