View source for Strongly countably complete topological space

A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306001.png" /> for which there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306002.png" /> of open coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306003.png" /> such that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306004.png" /> of closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306005.png" /> has a non-empty intersection whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306006.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306007.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306008.png" /> is a subset of some member of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306009.png" />.

Locally countably compact spaces and Čech-complete spaces are strongly countably complete. Every strongly countably complete space is a [[Baire space|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s13060010.png" /> implies joint continuity on a large subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s13060011.png" />, see [[Namioka space|Namioka space]]; [[Namioka theorem|Namioka theorem]]; [[Separate and joint continuity|Separate and joint continuity]]; or [[#References|[a2]]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Frolík,   "Baire spaces and some generalizations of complete metric spaces"  ''Czech. Math. J.'' , '''11'''  (1961)  pp. 237–248</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Namioka,   "Separate continuity and joint continuity"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 515–531</TD></TR></table>