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Hi I am very new to topology and was wondering how to solve the following problem :

Let X be an infinite set and τ a topology on X. If every infinite subset of X is in τ, prove that τ is the discrete topology.

I have trouble understanding why this would be true. It goes against my intuition of an infinite set and I don't quite understand how this topology would be discrete. As you could still pick any finite subset of X. Can anybody tell me why this true? Thanks in advance.

J. W. Tanner
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Jip Helsen
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It suffices to show that for every $x\in X$, $\{x\}\in\tau$. Since $X$ is infinite, we can take an infinite countable subset $E\subset X$. Let $E=\{a_n\}_{n\in\mathbb N}$. Then $A=\{x\}\cup\{a_{2n-1}\}_{n\in\mathbb N}$ and $B=\{x\}\cup\{a_{2n}\}_{n\in\mathbb N}$ are infinite subsets of $X$, so $A,B\in\tau$. Since $\tau$ is a topology, we have $A\cap B=\{x\}\in\tau$, which completes the proof.

J. W. Tanner
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Vivic
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  • thx I ll aprove asap – Jip Helsen Jul 07 '23 at 11:18
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    Does the first step require some form of the axiom of choice? See e.g. https://math.stackexchange.com/questions/201324/countable-subset-of-a-uncountable-set – Glenn Willen Jul 07 '23 at 19:56
  • It does require. You start taking $a_1\in X$ arbitrarialy. Since $X$ is infinite, $X-{a_1}$ is not empty, so we can choose $a_2\in X-{a_1}$. We continue this process recursively. – Vivic Jul 07 '23 at 21:07