Theorem 3

Let $f$ be a lower semi-continuous function on a non-empty quasi-compact space $X$. Then there is at least one point $a \in E$ such that $f(a) = \inf_{x \in X} f(x)$ (in other words, $f$ attains its greatest lower bound in $X$).

For each $k \in f(X)$, consider the set $A_k = \overline{f}([-\infty, k])$. These sets are non-empty and form a filter base on $X$; since they are closed by Proposition 1, they have at least one common point $a$ [axiom (C'') for quasi-compact spaces]. For each $x \in X$ we have therefore $f(a) \leq f(x)$, and the theorem follows.

(*) We recall (Set Theory, Chapter III, § 5, no. 5) that the characteristic function $\varphi_A$ of a subset $A$ of a set $X$ is the function defined on $X$ such that $\varphi_A(x) = 1$ for all $x \in A$, and $\varphi_A(x) = 0$ for all $x \in \complement_A$.