Is the cofinite topology on a countable set first countable? I know it is not first countable on R but I don't know the case of a countable set.
2026-03-25 22:02:51.1774476171
Is the cofinite topology on a countable set first countable?
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Not only is it first countable, it's second countable! That's because there are only countably many finite subsets of a countable set.
For instance, suppose the elements of the set are the naturals, since there would be a bijection between them. Then every finite subset could be paired with a different rational number, in a natural way. Say the rational number, in binary, with zero to the left of the decimal, and ones in the $n$-th place after the decimal, zeros elsewhere.
But, then we have an injection into the rationals, which are countable.