A compact Hausdorff space It is known that every finite space is compact. Then I am worried whether there exists a compact Hausdorff space $X$ with with ordinal of $X$ is $\omega_0$.
Does anyone know about it?

Consider the set formed by the sequence {1/n} union the point {0}. This is a closed and bounded subset of the real numbers so it's compact. By definition of a sequence it's countable. It's also Hausdorff since it's a subset of a Hausdorff space (the reals).
