Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$:
- Is it compact?
- Is it 2nd countable?
$\omega$ is the first infinite ordinal and $\omega_1$ is the first uncountable ordinal. The topology is the product topology.
I know that the topological space $[0,\omega_1)$ is not compact, leading me to conclude that the answer for (1) is 'no' using the cartesian product definition of two compact spaces, but here (at least) one of them fails to be compact. Anything pointing me in the right direction will be helpful.