Find a Topology where union of compact are non compact

Question

Is it possible find a non-Hausdoff topology where $A$ and $B$ are compact
but $A \cup B$ is non-compact? Thanks

Answer 1

I don't believe so. In particular, let $\{U_i\}_{i \in I}$ be an open cover of $A \cup B$. Notice that this is also an open cover of $A$ and $B$ by themselves. Since $A$ and $B$ are compact, we can find two finite subsets of $\{U_i\}_{i \in I}$, one of which covers $A$ and the other of which covers $B$. See where this is going?

On the other hand, you should consider whether the intersection of two compact sets is necessarily compact and whether this answer depends on Hausdorff-ness.

Spoiler:

Click here for my solution to this problem.

Answer 2

The union of two compact sets is always compact, whether Hausdorff or not. The easy way to see this: given any open cover of $A \cup B$, the union of a finite subset that covers $A$ and a finite subset that covers $B$ is a finite set that covers $A \cup B$.