What are these as topological spaces? A proof I'm reading is referring to these as counter examples and I can't figure out what they mean?
The answer is come from here. It is very interesting. So I copy it for you as an answer.
I find pictures to help. The idea here is that $\omega$ is a limit ordinal and tacking on the ordinal $1$ after it is fundamentally different:
The picture for $\omega$ has a curved edge which indicates that it is a limit ordinal opposed to being a successor ordinal. When we tack on $1$ to the right of $\omega$ we have this ordinal $\omega+1$ that contains a limit ordinal which is not something that occurs in $\omega$. This means that $\omega$ and $\omega+1$ can't be isomorphic.
These are ordinal spaces. Ordinals are linearly ordered sets which are also well-ordered, namely every subset has a least element (a generalization of the natural numbers with their ordering).
$\omega_1$ is the linear ordering that every initial segment is countable, but the whole set is uncountable. $\omega_1+1$ is the same order, but now with a maximum which is the unique point that has uncountably many points smaller than itself.
Linear orders generate a very nice topology with the order topology, which is what and how we usually topologize ordinals (I think that any other topology would have been explicitly mentioned).