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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".

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".

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".

edited for clarity
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T. Amdeberhan
T. Amdeberhan
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QUESTION.

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

In case there is, what would be an example of 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".

In case there is, what would be an example of 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".

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".

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Cauchy
Cauchy
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A topology on $\Bbb R$ where the compact sets are precisely the countable sets

In case there is, what would be an example of 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".