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Advanced Calculus: Difference between revisions
→Compact
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Intersection of infinitely many open sets can be closed. | Intersection of infinitely many open sets can be closed. | ||
===Compact=== | ===Compact=== | ||
Compactness is a generalization of closed and bounded.<br> | |||
;Definitions | |||
* (Sequence) A set is sequentially compact if for every sequence from the set, there exists a subsequence which converges to a point in the set. | |||
* (Topology) A set is compact if for every covering by infinitely many open sets, there exists a covering by a finite subset of the open sets. | |||
;Notes | |||
* A set is sequentially compact iff it is closed and bounded | |||
===Metric Space=== | ===Metric Space=== | ||