For every topological property $\mathcal{P}$, we define the class of $\mathcal{P}$-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 $\mathcal{P}$ for each $x\in X$. It is shown that every $\mathcal{P}$-approximable compact space has $\mathcal{P}$, if $\mathcal{P}$ 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...