Compact set, countably

A set $M$ in a topological space $X$ that as a subspace of this space is countably compact (cf. Compact space, countably). Countable compactness means that every sequence has an accumulation point, i.e. a point every neighbourhood of which contains infinitely many terms of the sequence.

A topological space $X$ is called sequentially compact if every sequence has a converging subsequence, i.e. if every sequence has a subsequence converging to some point of $X$ (cf. Sequentially-compact space).

A set $M$ in a topological space $X$ is called relatively (sequentially, countably) compact if its closure has the corresponding property.

A set $M$ in a topological space $X$ such that every infinite sequence $\{ x_i : i \in \mathbb{Z}\,,\ x_i \in M \}$ has a subsequence converging to some point $x_0$ of $X$ (respectively, has an accumulation point) could be called conditionally sequentially compact (respectively, conditionally countably compact).

Comments

In metric spaces and Banach spaces with the weak topology the notions of compactness, sequential compactness and countable compactness coincide.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)