Compactness of intersection of a compact set and an open set If $K \subset E_1 \cup E_2$, where $K$ is compact and $E_1, E_2$ are disjoint open subsets of a topological space, is $K \cap E_1$ compact? Is that always the case if $E_1, E_2$ are not disjoint?
I've seen other threads that have this, so I was wondering why the solution I thought of is incorrect:
Let $U_{\alpha}$ be an open covering of $K$. Then because $K$ is compact, there is a finite subcovering $U_1, U_2, \ldots, U_N \in U_{\alpha}$ that cover $K$. But then $U_1, \ldots, U_N, E_1$ is a finite collection of open sets that covers $K$ and $E_1$, so it covers $K \cap E_1$, and so $K \cap E_1$ must be compact.

Assuming that your space is Hausdorff $K\setminus E_2=K\cap E_1$ and $K\setminus E_2$ is compact. On second thoughts you don't need Hausdorff property!. 
