Are you saying that your space is with the topology ?

What do you think?

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So, does every point have a countable neighborhood base?is (X,T) First Countable space ?

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I mean, the idea is pretty straightforward. Think about "countable endpoints"is (X,T) Second Countable space ?

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Hint:is (X,T) compact space ?

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No hint on this one, I want you to try it for yourself.is (X,T) locally compact space ?

thanks