Topology:Ordinals space

[Amer]

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 ?

 

[Opalg]

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 ?

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]

can we treat [tex]\omega_1 [/tex] like infinity it is not a number and there is not an open set containing infinity without containing a natural number