Lemma 2

Let B be a topological space, and let K and L be quasi-compact subsets of B. Suppose that for every point $a$ of K and every point $b$ of L, there exist disjoint neighbourhoods of $a$ and $b$ in B. Then there exist two disjoint open sets U and V in B such that $K\subset U$ and $L\subset V$.

Let $a$ be a point of K. For every $b\in L$, let $U_{a,b}$ and $V_{a,b}$ be disjoint open neighbourhoods of $a$ and of $b$. For fixed $a$, the family $(V_{a,b})_{b\in L}$ is an open covering of L. Since this space is quasi-compact, there exists a finite family $T_a\subset L$ such that the union $V_b$ of the $V_{a,b}$ for $b\in T_a$ contains L. The intersection $U_a$ of the $U_{a,b}$ for $b\in T_a$ is an open neighbourhood of $a$, since $T_a$ is finite; moreover, $U_a$ and $V_a$ are disjoint. The $(U_a)_{a\in A}$ form an open covering of K. Since K is quasi-compact, there exists a finite family $S\subset K$ such that $U =\bigcup_{a\in S}U_a$ contains K. Then $V =\bigcap_{a\in S}V_a$ is an open subset of B which contains L. The lemma is proved.