View source for Separate and joint continuity

It follows from a property of the product topology that every [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201001.png" /> between topological spaces is separately continuous, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201002.png" /> is continuous with respect to each variable while the other variable is fixed [[#References|[a3]]]. It was observed by E. Heine [[#References|[a14]]], p. 15, that, in general, the converse does not hold (see also [[#References|[a11]]] for an account of early discoveries in this field).

Given  "nice"  topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201004.png" /> (cf. also [[Topological space|Topological space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201005.png" /> be a [[Metric space|metric space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201006.png" /> be separately continuous. Questions on separate and joint continuity are, among others, problems of the type:

the existence problem: Find the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201007.png" /> of points of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201008.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010010.png" /> are  "nice" , then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010011.png" /> is a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010012.png" />-subset (cf. [[Set of type F sigma(G delta)|Set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010014.png" />)]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010015.png" />. For example, every real-valued separately continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010016.png" /> is of the first Baire class (cf. also [[Baire classes|Baire classes]]), hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010017.png" /> is a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010018.png" />-subset.

There is also interest in a  "fibre"  version; it is the same as above, except now one looks for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010020.png" />, for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010021.png" />.

The characterization problem. Characterize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010022.png" /> as a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010023.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010024.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010025.png" /> is the complement of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010026.png" />-set contained in the product of two sets of the first Baire category [[#References|[a8]]].

The uniformization problem. Find out whether there is a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010027.png" />-subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010029.png" /> such that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010030.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010031.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010032.png" />, such a result was known already to R. Baire [[#References|[a1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010033.png" /> is a complete metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010034.png" /> is a compact metric space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010035.png" />, the uniformization problem was positively solved by H. Hahn [[#References|[a7]]]. I. Namioka [[#References|[a9]]] extended Hahn's result to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010036.png" /> being a regular, strongly countably complete space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010037.png" /> being a locally compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010038.png" />-compact space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010039.png" /> being a pseudo-metric space.

Following [[#References|[a4]]], one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010040.png" /> is a Namioka space if for any compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010041.png" />, any metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010042.png" /> and any separately continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010043.png" />, the uniformization problem can be positively solved. M. Talagrand [[#References|[a13]]] constructed an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010045.png" />-favourable space (hence, a Baire space) that is not Namioka. J. Saint Ramond [[#References|[a12]]] proved that separable Baire spaces are Namioka, Tikhonov Namioka spaces are Baire, while in the class of metric spaces, the set of Namioka spaces and the set of Baire spaces coincide.

A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010046.png" /> is a co-Namioka space if for any Baire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010047.png" />, any metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010048.png" /> and for any separably continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010049.png" />, the uniformization problem can be positively solved. For example, Corson-compact spaces are co–Namioka, whereas the Čech–Stone compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s12010050.png" /> is not. Many results in this direction have been obtained by R. Haydon, R.W. Hansell, J.E. Jain, J.P. Troallic, Namioka, and R. Pol (see [[#References|[a10]]] for a comprehensive exposition of this topic, organizing research in this field until the middle of the 1980s).

==Applications.==
R. Ellis [[#References|[a5]]], [[#References|[a6]]] showed that every locally compact semi-topological group (i.e., a group endowed with a topology for which the product is separately continuous) is a [[Topological group|topological group]]. Using methods of separate and joint continuity, A. Bouziad [[#References|[a2]]] extended Ellis' theorem to all Čech-analytic Baire semi-topological groups (hence, to all Čech-complete semi-topological groups).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Baire,   "Sur les fonctions des variables réelles"  ''Ann. Mat. Pura Appl.'' , '''3'''  (1899)  pp. 1–122</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bouziad,   "Every Čech-analytic Baire semitopological group is a topological group"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 953–959</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.L. Cauchy,   "Cours d'Analyse de l'Ecole Polytechnique"  (1821)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.P.R. Christensen,   "Joint continuity of separately continuous functions"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 455–462</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Ellis,   "A note on the continuity of the inverse"  ''Proc. Amer. Math. Soc.'' , '''8'''  (1957)  pp. 372–373</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Ellis,   "Locally compact transformation groups"  ''Duke Math. J.'' , '''24'''  (1957)  pp. 119–125</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Hahn,   "Reelle Funktionen" , Akad. Verlag  (1932)  pp. 325–338</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Kershner,   "The continuity of functions of many variables"  ''Trans. Amer. Math. Soc.'' , '''53'''  (1943)  pp. 83–100</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I. Namioka,   "Separate and joint continuity"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 515–531</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Z. Piotrowski,   "Separate and joint continuity"  ''Real Anal. Exch.'' , '''11'''  (1985-86)  pp. 293–322</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Z. Piotrowski,   "The genesis of separate versus joint continuity"  ''Tatra Mtn. Math. Publ.'' , '''8'''  (1996)  pp. 113–126</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Saint Raymond,   "Jeux topologiques et espaces de Namioka"  ''Proc. Amer. Math. Soc.'' , '''87'''  (1983)  pp. 499–504</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  M. Talagrand,   "Espaces de Baire et espaces de Namioka"  ''Math. Ann.'' , '''270'''  (1985)  pp. 159–164</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J. Thomae,   "Abriss einer Theorie der complexen Funktionen" , Louis Nebert Verlag  (1873)  (Edition: Second)</TD></TR></table>