Let $A$ and $B$ be disjoint compact subsets of a Hausdorff space $(X,τ)$. Prove that there exist disjoint open sets $G$ and $H$ such that $A⊆G$ and $B⊆H$.
This one was getting a bit messy for me, but my idea is to first fix a point $x$ in $B$ and then apply Hausdorff property on points of $A$ so for all points an in $A$ with $x$ in $B$ we can find disjoint open sets containing them. We can say $U_a$ and $V_a$ be the set of all open sets we just created such that $U_a$ and $V_a$ are disjoint. We note that $U_a$ is an open cover for $A$ and we can pick out a finite subcover.
Then we select another point in $B$ and repeat the same process and do this for every single point in $B$. Then we have a family of finite subcovers each corresponding to each point in $B$. We can take the intersection and call it $W$ and let that be the open set that contains $A$.
Then the next step would be to repeat everything above by fixing a point in $A$ and cycling through points in $B$ then fixing another point in $A$ and create a family of open subcovers to cover $B$ and intersect them all and call that $G$.
Now I believe $A$ is contained in $W$ and $B$ is in $G$ and $W$ and $G$ are disjoint. A concern I have is that while each subcover is finite, the family of subcover may be uncountable and the intersection of uncountable open sets are not always open.
