Abstract: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, \v Cech-complete, paracompact subspace; moreover, if $X$ is linearly ordered, then $X$ contains a dense metrizable subspace.
Keywords: $\Cal P$-approximable space, Lindel\"of $\Sigma $-space, compact, metrizable, $C$-closed, sequential, linearly ordered
AMS Subject Classification: 54D20, 54D30, 54E35, 54F05