Characterizations of p-Spaces

Abstract

The concept of p-space is quite recent. It was introduced by Arhangel'skii [2]. The definition of p-space given in [2] involves compactification of the space. In view of the interesting properties of p-spaces obtained in [2], Alexadroff [1] suggested a problem of finding a direct intrinsic definition (without appeal to compactification). The main aim of this note is to answer the above problem.I am grateful to Dr. S. K. Kaul for his comments.

On $D$-paracompact $p$- and $\Sigma$-spaces

“…Since D-paracompact spaces are submetacompact, the arguments of [8, Theorem 3.19 and 3.21] can apply to get the equivalence of (i), (ii) and (iii). If we again note the remark in [8, p. 442], the discussion of [13] holds true for regular spaces, so that we have the equivalence of (iv) and (iii). $(iii)\rightarrow(v)$ : Let $\mathscr{U}$ be an open cover of $X$ and let $\{\mathscr{G}_{n} : n\in N\}$ be a strict p-sequence for $X$ satisfying the following:…”

Moore Spaces, Semi-Metric Spaces and Continuous Mappings Connected with Them

“…Hence F is a Moore space and the lemma is proved. In view of Theorem 3.1 in [6] we take the definition of £-space as follows: (c) given i and x in X, there is a j such that for any y in st(x, Yf) there is fc" such that st(y, V ky ) C st(x,^f); and (d)if{F\F G ^] is a family of closed sets with finite intersection property and there is x in X such that for each i some F in J^~ is contained in st(x,^" z ) then C\{F\F G ^H ^ 0. Now using a proof analogous to Theorem 3.4, one can show that there exists a perfect mapping of X onto a Moore space Y.…”