# Strongly countably complete topological space

A topological space for which there is a sequence of open coverings of such that a sequence of closed subsets of has a non-empty intersection whenever for all and each is a subset of some member of .

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 implies joint continuity on a large subset of , 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=17711