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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]]).
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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" />.
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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]]).
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Let $X$ be a regular, strongly countably complete [[Topological space|topological space]] (cf. also [[Strongly countably complete topological space|Strongly countably complete topological space]]), let $Y$ be a locally compact and $\sigma$-compact space (cf. also [[Compact space|Compact space]]) and let $Z$ be a [[Pseudo-metric space|pseudo-metric space]]. In 1974, I. Namioka [[#References|[a7]]] proved that for every separately continuous function $f : X \times Y \rightarrow Z$ there is a dense $G _ { \delta }$-subset $A$ of $X$ such that the set $A \times Y$ is contained in $C ( f )$, the set of points of continuity of $f$ (cf. also [[Set of type F sigma(G delta)|Set of type $F _ { \sigma }$ ($G _ { \delta }$)]]; [[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]]]).
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The original proof of this theorem starts with an interesting reduction to the case when $Y$ 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 $O _ { \mathcal{E} }$ is the union of all open subsets $0$ of $X \times Y$ such that $\operatorname{diam}f ( 0 ) \leq \varepsilon$, the set $A _ { \varepsilon } = \{ x : \{ x \} \times Y \subset O _ { \varepsilon } \}$ is dense in $X$.
  
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.
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For $X = Y = Z = \bf R$ (the real numbers), such a result was known already to R. Baire [[#References|[a2]]] (cf. [[Separate and joint continuity|Separate and joint continuity]]).
  
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]]].
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If $X$ is complete metric, $Y$ is compact metric and $Z = \mathbf{R}$, Namioka's theorem was shown by H. Hahn [[#References|[a6]]] (see also [[#References|[a11]]]).
  
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.
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The question whether the completeness of $Y$ 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 $Y$. In fact, let $X = [ 0,1 ]$, $Y = \cup _ { \alpha \in [ 0,1 ] } Y _ { \alpha }$, where $Y _ { \alpha } = [ 0,1 ]$ and $\cup$ denotes the free union of, in fact, $c$ many copies of $[ 0,1 ]$. Let $f : X \times Y \rightarrow \bf R$ be separately continuous on every  "square" $X \times Y _ { \alpha }$ and having a point of discontinuity along the line $x = \alpha$. Then, clearly, the set $A$ mentioned in Namioka's theorem is empty.
  
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]]].
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Answering a problem of Namioka, it was shown [[#References|[a12]]] that Namioka's theorem fails for all Baire spaces $X$ (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 $\sigma$-$\beta$-defavourable spaces [[#References|[a3]]], [[#References|[a10]]] and for Baire spaces having dense subsets that are countable unions of $\bf K$-analytic subsets [[#References|[a13]]].
 +
 
 +
The importance of Namioka's theorem lies in the fact that both $X$ and $Y$ are neither metrizable nor having any kind of countability of basis.
 +
 
 +
If $Y$ has a countable base, then Namioka's theorem holds for all Baire spaces $X$, see [[#References|[a4]]] and [[#References|[a9]]].
  
 
For further information, see [[Namioka space|Namioka space]].
 
For further information, see [[Namioka space|Namioka space]].
  
 
====References====
 
====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>
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<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 ($6_\beta$)  (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>

Latest revision as of 17:02, 1 July 2020

Let $X$ be a regular, strongly countably complete topological space (cf. also Strongly countably complete topological space), let $Y$ be a locally compact and $\sigma$-compact space (cf. also Compact space) and let $Z$ be a pseudo-metric space. In 1974, I. Namioka [a7] proved that for every separately continuous function $f : X \times Y \rightarrow Z$ there is a dense $G _ { \delta }$-subset $A$ of $X$ such that the set $A \times Y$ is contained in $C ( f )$, the set of points of continuity of $f$ (cf. also Set of type $F _ { \sigma }$ ($G _ { \delta }$); Separate and joint continuity).

The original proof of this theorem starts with an interesting reduction to the case when $Y$ 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 $O _ { \mathcal{E} }$ is the union of all open subsets $0$ of $X \times Y$ such that $\operatorname{diam}f ( 0 ) \leq \varepsilon$, the set $A _ { \varepsilon } = \{ x : \{ x \} \times Y \subset O _ { \varepsilon } \}$ is dense in $X$.

For $X = Y = Z = \bf R$ (the real numbers), such a result was known already to R. Baire [a2] (cf. Separate and joint continuity).

If $X$ is complete metric, $Y$ is compact metric and $Z = \mathbf{R}$, Namioka's theorem was shown by H. Hahn [a6] (see also [a11]).

The question whether the completeness of $Y$ suffices in Hahn's result was asked, independently, in [a1] and [a5]. The following example, due to J.B. Brown [a8] shows that completeness does not suffice and proves the necessity of compactness of $Y$. In fact, let $X = [ 0,1 ]$, $Y = \cup _ { \alpha \in [ 0,1 ] } Y _ { \alpha }$, where $Y _ { \alpha } = [ 0,1 ]$ and $\cup$ denotes the free union of, in fact, $c$ many copies of $[ 0,1 ]$. Let $f : X \times Y \rightarrow \bf R$ be separately continuous on every "square" $X \times Y _ { \alpha }$ and having a point of discontinuity along the line $x = \alpha$. Then, clearly, the set $A$ mentioned in Namioka's theorem is empty.

Answering a problem of Namioka, it was shown [a12] that Namioka's theorem fails for all Baire spaces $X$ (cf. also Baire space). Still, the theorem holds for certain Banach–Mazur game-defined spaces (cf. also Banach–Mazur game), namely for $\sigma$-$\beta$-defavourable spaces [a3], [a10] and for Baire spaces having dense subsets that are countable unions of $\bf K$-analytic subsets [a13].

The importance of Namioka's theorem lies in the fact that both $X$ and $Y$ are neither metrizable nor having any kind of countability of basis.

If $Y$ has a countable base, then Namioka's theorem holds for all Baire spaces $X$, see [a4] and [a9].

For further information, see Namioka space.

References

[a1] A. Alexiewicz, W. Orlicz, "Sur la continuité et la classification de Baire des fonctions abstraites" Fundam. Math. , 35 (1948) pp. 105–126
[a2] R. Baire, "Sur les fonctions des variables réelles" Ann. Mat. Pura Appl. , 3 (1899) pp. 1–122
[a3] A. Bouziad, "Jeux topologiques et point de continuité d'une application séparément continue" C.R. Acad. Sci. Paris , 310 (1990) pp. 359–361
[a4] J. Calbrix, J.P. Troallic, "Applications séparément continue" C.R. Acad. Sci. Paris Sér. A , 288 (1979) pp. 647–648
[a5] J.P.R. Christensen, "Joint continuity of separately continuous functions" Proc. Amer. Math. Soc. , 82 (1981) pp. 455–461
[a6] H. Hahn, "Reelle Funktionen" , Leipzig (1932) pp. 325–338
[a7] I. Namioka, "Separate and joint continuity" Pacific J. Math. , 51 (1974) pp. 515–531
[a8] Z. Piotrowski, "Separate and joint continuity" Real Analysis Exchange , 11 (1985/86) pp. 293–322
[a9] Z. Piotrowski, "Topics in separate and joint continuity" in preparation (2001)
[a10] J. Saint-Raymond, "Jeux topologiques et espaces de Namioka" Proc. Amer. Math. Soc. , 87 (1983) pp. 499–504
[a11] R. Sikorski, "Funkcje rzeczywiste" , I , PWN (1958) pp. 172; Problem ($6_\beta$) (In Polish)
[a12] M. Talagrand, "Propriété de Baire et propriété de Namioka" Math. Ann. , 270 (1985) pp. 159–174
[a13] G. Debs, "Points de continuité d'une fonction séparément continue" Proc. Amer. Math. Soc. , 97 (1986) pp. 167–176
How to Cite This Entry:
Namioka theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Namioka_theorem&oldid=11989
This article was adapted from an original article by Z. Piotrowski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article