Topology:Ordinals space

Amer

Active member
Is the subset of the ordinals space
$$\omega = [1, \omega_1 ]$$ first countable
It has discrete topology so it is first countable, countable local base at each point

is it correct ?

Opalg

MHB Oldtimer
Staff member
Is the subset of the ordinals space
$$\omega = [1, \omega_1 ]$$ first countable
It has discrete topology so it is first countable, countable local base at each point

is it correct ?
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.

Amer

Active member
can we treat $$\omega_1$$ like infinity it is not a number and there is not an open set containing infinity without containing a natural number