This space does not have the discrete topology, because every limit ordinal fails to be a discrete point. In fact, this space is not first countable because the endpoint $\omega_1$ does not have a countable local base. See http://en.wikipedia.org/wiki/First-countable_space#Examples_and_counterexamples.Is the subset of the ordinals space
[tex]\omega = [1, \omega_1 ] [/tex] first countable
It has discrete topology so it is first countable, countable local base at each point
is it correct ?