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Topology:Ordinals space

Amer

Active member
Mar 1, 2012
275
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,721
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

Active member
Mar 1, 2012
275
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