View source for Namioka theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300201.png" /> be a regular, strongly countably complete [[Topological space|topological space]] (cf. also [[Strongly countably complete topological space|Strongly countably complete topological space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300202.png" /> be a locally compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300203.png" />-compact space (cf. also [[Compact space|Compact space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300204.png" /> be a [[Pseudo-metric space|pseudo-metric space]]. In 1974, I. Namioka [[#References|[a7]]] proved that for every separately continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300205.png" /> there is a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300206.png" />-subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300207.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300208.png" /> such that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n1300209.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002010.png" />, the set of points of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002011.png" /> (cf. also [[Set of type F sigma(G delta)|Set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002013.png" />)]]; [[Separate and joint continuity|Separate and joint continuity]]).

The original proof of this theorem starts with an interesting reduction to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002014.png" /> is compact. Next, using purely topological methods, such as, e.g., the Arkhangel'skii–Frolík covering theorem and Kuratowski's theorem on closed projections, Namioka shows that, given that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002015.png" /> is the union of all open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002018.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002019.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002020.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002021.png" /> (the real numbers), such a result was known already to R. Baire [[#References|[a2]]] (cf. [[Separate and joint continuity|Separate and joint continuity]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002022.png" /> is complete metric, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002023.png" /> is compact metric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002024.png" />, Namioka's theorem was shown by H. Hahn [[#References|[a6]]] (see also [[#References|[a11]]]).

The question whether the completeness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002025.png" /> suffices in Hahn's result was asked, independently, in [[#References|[a1]]] and [[#References|[a5]]]. The following example, due to J.B. Brown [[#References|[a8]]] shows that completeness does not suffice and proves the necessity of compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002026.png" />. In fact, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002030.png" /> denotes the free union of, in fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002031.png" /> many copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002032.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002033.png" /> be separately continuous on every  "square"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002034.png" /> and having a point of discontinuity along the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002035.png" />. Then, clearly, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002036.png" /> mentioned in Namioka's theorem is empty.

Answering a problem of Namioka, it was shown [[#References|[a12]]] that Namioka's theorem fails for all Baire spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002037.png" /> (cf. also [[Baire space|Baire space]]). Still, the theorem holds for certain Banach–Mazur game-defined spaces (cf. also [[Banach–Mazur game|Banach–Mazur game]]), namely for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002040.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002041.png" />-defavourable spaces [[#References|[a3]]], [[#References|[a10]]] and for Baire spaces having dense subsets that are countable unions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002042.png" />-analytic subsets [[#References|[a13]]].

The importance of Namioka's theorem lies in the fact that both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002044.png" /> are neither metrizable nor having any kind of countability of basis.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002045.png" /> has a countable base, then Namioka's theorem holds for all Baire spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002046.png" />, see [[#References|[a4]]] and [[#References|[a9]]].

For further information, see [[Namioka space|Namioka space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Alexiewicz,   W. Orlicz,   "Sur la continuité et la classification de Baire des fonctions abstraites"  ''Fundam. Math.'' , '''35'''  (1948)  pp. 105–126</TD></TR><TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top">  A. Bouziad,   "Jeux topologiques et point de continuité d'une application séparément continue"  ''C.R. Acad. Sci. Paris'' , '''310'''  (1990)  pp. 359–361</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Calbrix,   J.P. Troallic,   "Applications séparément continue"  ''C.R. Acad. Sci. Paris Sér. A'' , '''288'''  (1979)  pp. 647–648</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.P.R. Christensen,   "Joint continuity of separately continuous functions"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 455–461</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Hahn,   "Reelle Funktionen" , Leipzig  (1932)  pp. 325–338</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I. Namioka,   "Separate and joint continuity"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 515–531</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Z. Piotrowski,   "Separate and joint continuity"  ''Real Analysis Exchange'' , '''11'''  (1985/86)  pp. 293–322</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  Z. Piotrowski,   "Topics in separate and joint continuity"  ''in preparation''  (2001)</TD></TR><TR><TD valign="top">[a10]</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">[a11]</TD> <TD valign="top">  R. Sikorski,   "Funkcje rzeczywiste" , '''I''' , PWN  (1958)  pp. 172; Problem (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130020/n13002047.png" />)  (In Polish)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Talagrand,   "Propriété de Baire et propriété de Namioka"  ''Math. Ann.'' , '''270'''  (1985)  pp. 159–174</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  G. Debs,   "Points de continuité d'une fonction séparément continue"  ''Proc. Amer. Math. Soc.'' , '''97'''  (1986)  pp. 167–176</TD></TR></table>