$X$ compact Hausdorff implies $x\mapsto \mathfrak{m}_x$ is a homeomorphism

Question

Let $X$ be a compact Hausdorff space. Denote by $\mathfrak m_x$ the prime ideal of $C(X)$ comprised of functions vanishing at $x$. Topologize $\operatorname{MaxSpec}C(X)$ with the initial topology from $X$. I want to prove the map $x\mapsto \mathfrak m_x$ is a homeomorphism. I know it's bijective. Its inverse is continuous as a map $\operatorname{MaxSpec}C(X)\rightarrow X$, but why is it the assignment $x\mapsto \mathfrak m_x$ continuous?

Answer 1

Its inverse is continuous and closed i.e it sends closed to closed hence it sends open to open (it is bijective). This mean that your map is continuous.

Closed because the two spaces are compacts, and a closed subset is compact, so its image is compact, in particular closed (Hausdorff).