1
vote
1answer
111 views

Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
3
votes
1answer
70 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
1
vote
2answers
86 views

Is $\omega_{\alpha}$ sequentially compact?

For an ordinal $\alpha \geq 2$, let $\omega_{\alpha}$ be as defined here. It is easy to show that $\omega_{\alpha}$ is limit point compact, but is it sequentially compact?
12
votes
3answers
767 views

Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...