Countable but not First Countable Space

Asked May 29 '23 at 12:39

Active May 29 '23 at 12:39

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I came across this question in Chapter 5 of Biglist. I have no idea how to grab a handle on the question.

Construct a Topological space $(X, \mathcal{T})$ which is countable (i.e., $X$ is countable) but not First countable.

I cannot conceive of any topological space which is countable but not first countable.

Any help is greatly appreciated.

asked May 29 '23 at 12:39

Ryukendo Dey

Consider the one-point compactification of the rationals. – balddraz May 29 '23 at 13:04
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Does this answer your question? Example of a topological space that has countably (but not finitely) many elements, is regular, but not 2nd countable Note that "first countable" and "second countable" are equivalent for spaces with countably many points. – bof May 29 '23 at 16:10
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A few examples: https://topology.pi-base.org/spaces?q=Countable%2B~Second%20Countable – Steven Clontz May 29 '23 at 22:36

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