>**QUESTION.** 

>In there a topology on $\Bbb R$ where the compact subsets are precisely the countable subsets? 

I am trying to create a counterexample to a certain claim, and I found that what I need is a topology of this kind. I thought hard about this and did quite a lot of searching, but could not find something relevant. Thank you in advance.

Note: "countable" includes "finite".