$\Cal P$-approximable compact spaces

Tkačenko, Michael G.

Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 583-595.

Résumé

For every topological property $\Cal P$, we define the class of $\Cal P$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the ``section'' $X(x,\gamma )= \bigcap \{F\in \gamma :x\in F\}$ has the property $\Cal P$ for each $x\in X$. It is shown that every $\Cal P$-approximable compact space has $\Cal P$, if $\Cal P$ is one of the following properties: countable tightness, $\aleph _0$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or $2^{\aleph _0}2^{\aleph _1}$). Metrizable-approximable spaces are studied: every compact space in this class has a dense, Čech-complete, paracompact subspace; moreover, if $X$ is linearly ordered, then $X$ contains a dense metrizable subspace.

Détail

Classification : 54A20, 54A25, 54A35, 54B05, 54B10, 54D20, 54D30, 54D55, 54E35, 54F05
Keywords: $\Cal P$-approximable space; Lindelöf $\Sigma $-space; compact; metrizable; $C$-closed; sequential; linearly ordered