Example 2

Let $X$ be a set, and give $X$ the topology in which the closed sets are $X$ and all finite subsets of $X$ [this set of subsets clearly satisfies axioms $(O'_1)$ and $(O'_{II})$ of $§ 1$, no. 4]. The topological space so defined is quasi-compact. For if $(F_i)_{i \in I}$ is a family of closed subsets of $X$ with empty intersection, then $F_\alpha$ is finite for some $\alpha \in I$. Let $a_k$ ($1 \leq k \leq n$) be the elements of $F_\alpha$; then by hypothesis for each index $k$ there is an index $i_k \in I$ such that $a_k \notin F_{i_k}$; the intersection of the $F_{i_k}$ ($1 \leq k \leq n$) and $F_\alpha$ is therefore empty, whence axiom $(C'')$ is satisfied. If $X$ is infinite it is not Hausdorff.