Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

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Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.
{Ca}aea be a collection of subsets of X.
Cg. The collection C is an open cover
Definition. Let A be a subset of X and let C =
Then C is a cover of A if and only if A C Ugea
of A if and only if C is a cover of A and each Ca is open. A subcover C' of a cover C of
A is a subcollection of C whose elements form a cover of A.
For instance, the open sets {(-n,n)}nen form an open cover of R. A subcover of
this cover is {(-n,n)}n>5, because these sets still cover all of R.
Definition. A space X is compact if and only if every open cover of X has a finite
subcover.
Transcribed Image Text:Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point. {Ca}aea be a collection of subsets of X. Cg. The collection C is an open cover Definition. Let A be a subset of X and let C = Then C is a cover of A if and only if A C Ugea of A if and only if C is a cover of A and each Ca is open. A subcover C' of a cover C of A is a subcollection of C whose elements form a cover of A. For instance, the open sets {(-n,n)}nen form an open cover of R. A subcover of this cover is {(-n,n)}n>5, because these sets still cover all of R. Definition. A space X is compact if and only if every open cover of X has a finite subcover.
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