# Strongly countably complete topological space

A topological space $X$ for which there is a sequence $\{\mathcal A_i\}$ of open coverings of $X$ such that a sequence $\{F_i\}$ of closed subsets of $X$ has a non-empty intersection whenever $F_i\supset F_{i+1}$ for all $i$ and each $F_i$ is a subset of some member of $\mathcal A_i$.

Locally countably compact spaces and Čech-complete spaces are strongly countably complete. Every strongly countably complete space is a Baire space (but not vice versa).

This rather technical notion plays an important role in questions whether separate continuity of a mapping on a product $X\times Y$ implies joint continuity on a large subset of $X\times Y$, see Namioka space; Namioka theorem; Separate and joint continuity; or [a2].

Strongly countably complete topological spaces were introduced by Z. Frolík, [a1].

#### References

[a1] | Z. Frolík, "Baire spaces and some generalizations of complete metric spaces" Czech. Math. J. , 11 (1961) pp. 237–248 |

[a2] | I. Namioka, "Separate continuity and joint continuity" Pacific J. Math. , 51 (1974) pp. 515–531 |

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Strongly countably complete topological space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Strongly_countably_complete_topological_space&oldid=32427