Math 2.0

General topology Definition

Sequential compactness

Builds on Topological space

Definition

Let XX be a topological space and let K⊆XK \subseteq X. The set KK is sequentially compact when each sequence (xn)(x_n) of points of KK has a subsequence (xnk)(x_{n_k}) that converges to a point of KK. The space XX is sequentially compact when XX itself is a sequentially compact set.

Remarks

Here a sequence (yk)(y_k) converges to yy when each open set that contains yy contains yky_k for all large kk. In a metric space, this is the usual convergence. The interval [0,1][0, 1] is sequentially compact by the Bolzano–Weierstrass theorem. The real line is not: the sequence xn=nx_n = n has no convergent subsequence. For metric spaces, compactness and sequential compactness agree. In general topological spaces, neither notion implies the other.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionTopological space, definition, no evidence levelTopological spacedefinitionSequential compactness, definition, no evidence levelSequential compactnessdefinition
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  15. Union of a family of sets definition
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  17. Sequential compactness definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Sequentially_compact_space, lead section
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DefinesSequential compactness
UsesTopological space

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