A compact Hausdorff space

Question

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?

Answer 1

Every well order with a maximum in the order topology is compact. So take any well order of cardinal $\aleph_{0}$, add a maximum if it doesn't have one and you got your space.

(Every order topology is also $T_{2}$, in fact $T_{4}$)

Answer 2

Think of the following: $\Bbb N \cup \{\infty\}$, where $\infty$ is an abstract element defined by the property that $n \leq \infty \space \forall n \in \Bbb N$.

Answer 3

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).