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A set where every sequence has a convergent subsequence with limit in the set.
A subset of a topological space is sequentially compact if every sequence in has a subsequence that converges in to a point of .
In metric spaces, sequential compactness is equivalent to compactness. In general topological spaces, neither property implies the other without additional hypotheses.
A topological space is a pair , where is a set and is a collection of subsets of , satisfying:
The collection is the topology of the space.
A subsequence of a sequence is a sequence of the form , where is a strictly increasing sequence of indices.
Subsequences are a standard way to isolate “partial” limiting behavior of a sequence; they are used in defining limit superior and limit inferior. The monotone subsequence lemma guarantees monotone subsequences under mild hypotheses.
A sequence in a topological space converges to if, for every neighborhood of , there is such that for all .
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that