View source for Separate and joint continuity

It follows from a property of the [[product topology]] that every [[continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201001.png" /> between [[topological space]]s 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" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120100/s1201005.png" /> be a [[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 $F_\sigma$ ($G_\delta$)]]) 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]]), 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 topological 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 space|separable]] [[Baire space]]s are Namioka, Namioka [[Tikhonov space]]s 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]]. 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>