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Keywords:
$\Cal P$-approximable space; Lindelöf $\Sigma$-space; compact; metrizable; $C$-closed; sequential; linearly ordered
Summary:
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.
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