A topology on a countable set may be uncountable, such as discrete topology on natural numbers. Although that topology has uncountably many set, it is first countable, indeed, second countable. I wonder that is there a topology on natural numbers being not first countable?
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Note that a countable space is first countable if and only if it is second countable. – freakish Apr 09 '21 at 11:51
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My answer is not Are countable topological spaces second-countable?. Unfortinately, the theorem you write "a countable space is first countable if and only if it is second countable" doesn't answer for the question. – Dr. Ufuk Kaya Apr 09 '21 at 12:11
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The answer says that there is a non second-countable topology on $\mathbb{N}$. And thus it is also non first-countable based on the theorem. Everything you need is there. – freakish Apr 09 '21 at 13:13
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Thank you for your attention. – Dr. Ufuk Kaya Apr 09 '21 at 13:35