With out the axiom of choice is the following true: If a topology on $X$ contains every infinite subset of $X$, then this is the discrete topology.

Asked Mar 20 '15 at 17:33

Active Mar 20 '15 at 18:01

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Is there a way to prove the following without having to use the axiom of choice or its countable version?

Let $X$ be an infinite set and $T$ be a topology on $X$. If $T$ contains every infinite subset of $X$, then $T$ is the discrete topology.

edited Mar 20 '15 at 18:01

asked Mar 20 '15 at 17:33

user220502

2

The answer is no and this is a possible duplicate of these question and answer here1 (especially the answer), here2, . If this does not answer your question you should edit yours to make it more apparent that it doesn't answer the question. – Mar 20 '15 at 17:44

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