On Spaces and ω-Mappings

Abstract

This note is closely related, as far as methods are concerned, to [3]. In [3] Ponomarev established “In order for a regular space X to be Lindelöf, it is necessary and sufficient that for each open covering ω of the space X there exists an ω-mapping ƒ:X → Y onto some separable metric space Y.” It is the purpose of this note to show that if the word “countable (or finite)” is inserted in the proper place we can obtain an analogous characterization for normal countably paracompact (or normal) spaces.

On m-Paracompact Spaces and ω-Mappings

“…The purpose of this note is to present a characterization of m-paracompact normal spaces in terms of co-mappings into m-separable metric spaces; this result is almost contained in Morita's original paper [2] and is implicitly contained in Shapiro's thesis paper [5]. This result is a natural generalization of the well-known Katetov-Ponomarev characterization of paracompact spaces (see [4]), and a special case of it (when m= K 0 ) was recently discovered by Pareek [3].…”

“…If m is an infinite cardinal number, then m-separable metric spaces are defined in [5], and m-paracompact normal spaces are defined in [2]. For the concept of normal covers, see [7], and for the term co-mapping, see [3] or [4]. Proof.…”