What is the topology defined on $[0, \omega_1)$ it and how do we define its basis?

Question

The set $[0, \omega_1)$ is a countably compact topological space. How do we prove it to be a topological space. How does its basis elements look like?

Answer 1

The set is the first uncountable ordinal which is a linearly ordered set, of size $\aleph_1$, and whose elements are all countable ordinals (so it starts with $0,1,2,\ldots,\omega, \omega+1, \omega+2, \ldots, \omega+\omega,\ldots$ and the order is a well-order (every non-empty subset has a minimum). The topology on this set is the standard order topology, which in this case is: $0$ and every successor ordinal $\alpha+1$ is an isolated point, and if $\alpha$ is a limit ordinal, a local base are all sets $(\beta,\alpha]$ where $\beta < \alpha$. All subsets of the form $(\alpha, \rightarrow)$ are also open.

A typical feature of $\omega_1$ is that every countable sequence $(\alpha_n)_n$ has a supremum in $[0,\omega_1)$. This is what causes it to be countably compact, but not compact. It's a commonly used counterexample, not only in topology but also in measure theory e.g.