finite-closed topology

[chiyui]

Hi everyone,

 

I'm a beginner of topology and I'm doing a proof with the following details:

 

Let T be the finite-closed topology on a set X. If T is also the discrete topology, prove that the set X is finite.

 

I write my proof as follows:

 

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Proof:

 

Since T is the discrete topology, all singleton set {x} are in T. Hence, every subset of X is in T.

Since T is finite-closed, every sets in T must have finite complement with respect to X. But since every subset of X is in T, every subsets of X must have finite complement w.r.t. X. Hence, X is finite.

 

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May anyone commment on if my proof (as follows) is correct or not? Thanks a lot.