I'm struggling with a topology question and I'm having a hard time proving the following: If I know that $X$ is a compact Hausdorff space, $A$ is a closed subspace and $X/A$ denotes the quotient space of $X$ which identifies $A$ to a single point.
Then I'd like to show, that $X/A$ is a compact Hausdorff space.
So far I have that since $A$ is a closed subspace of a compact space $X$ then $A$ is compact as well.
Since $A$ is a subspace of a Hausdorff space then $A$ is a Hausdorff space as well.
Since $X/A$ is the quotient space of $X$, which identifies $A$ to a single point, then by the definition of a quotient space (Munkres p. 137) then $X/A$ is a partition of $X$ into disjoint subsets whose union is $X$. Also there exist a surjective map $p:X \to X/A$ that carries each point of $X$ to the the element of $X/A$ containing it. Since $p$ is a quotient map this is equivalent to saying that $p$ is continuous.
And now I can't seem to get any further. How do I prove that $X/A$ is compact and Hausdorff?