On $p$-spaces and their continuous maps

N. V. Velichko

Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 35-46

Résumé

The following theorems are the main results of this paper. Theorem 1. Let $f\colon X\to Y$ be a closed mapping of the weakly paracompact $p$-space $X$. In order that the space $Y$ be weakly paracompact and plumed, it is necessary and sufficient that the mapping $f$ be peripherally bicompact. \smallskip Theorem 2. {\it Let $f\colon X\to Y$ be a closed mapping of a weakly paracompact $p$-space $X$. Then $Y=Y_0\cup Y_1,$ where the set $Y_1$ is $\sigma$-discrete in $Y$ and the set $f^{-1}y$ is bicompact for each point $y\in Y_0$.} An example is constructed of a weakly paracompact, locally compact, $\sigma$-paracompact space which is not normal and which cannot be mapped perfectly onto a space with a refining sequence of coverings. Bibliography: 22 titles.