Proof That Sequential Compactness But Not Countable Compactness is a Topological Property

Theorem

Sequential compactness is a topological property but countable compactness is not.

Proof

Letandbe homeomorphic topological spaces. Then a homeomorphismfromtoexists.

Supposeis sequentially compact. Letbe a sequence inthen sinceis one to one and onto,is a sequence inSinceis sequentially compactcontains a subsequencewhich converges to a point

Since f is continuous,

Thuscontains a subsequencewhich converges to a pointandis sequentially comp-act.

To prove countable compactness is not, letbe an infinite subset ofthenis an infinite subset ofis countably compact hencehas an accumulation pointHencehas an accumulation pointandis countably compact.

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