locally pseudoregular compact and open set as union of open sets with compact supersets

Let \((X, \mathcal{T})\) be a locally pseudoregular compact topological space. Then for each subset \(O \subseteq X\) there exists an open cover \(O_i\) with associated compact sets \(O_i \subseteq C_i\), such that

\begin{align*} O =& \bigcup_{i \in I} C_i \\ =& \bigcup_{i \in I} \mathrm{int}(C_i) \end{align*}

see interior

for \(a \in O\), choose \(C_a\) as compact neighbourhood. Then

\begin{align*} \bigcup_{a \in O} C_a \end{align*}

is the desired covering

Note, that

\begin{align*} \bigcup_{a \in O} \mathrm{int}(C_a) \supseteq& \bigcup_{a \in O} \{a\} =& O \end{align*}