# 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) |

**How to Cite This Entry:**

Compact set, countably.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Compact_set,_countably&oldid=34252