$\Cal P$-approximable compact spaces

Tkachenko, Mihail G.

Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991), p. 583-595 / Harvested from Czech Digital Mathematics Library

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

Publié le : 1991-01-01

MR 1159804 | Zbl 0766.54019

Classification:  54A20,  54A25,  54A35,  54B05,  54B10,  54D20,  54D30,  54D55,  54E35,  54F05

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