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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300101.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/n130010/n1300102.png" /> be a locally compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300103.png" />-compact space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300104.png" /> be a [[Pseudo-metric space|pseudo-metric space]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300105.png" /> be an arbitrary separately continuous function (cf. also [[Separate and joint continuity|Separate and joint continuity]]).
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Let $X$ be a regular, strongly countably complete [[topological space]] (cf. also [[Strongly countably complete topological space]]), let $Y$ be a [[Locally compact space|locally compact]] and $\sigma$-compact space, let $Z$ be a [[pseudo-metric space]], and let $f$ be an arbitrary separately continuous function (cf. also [[Separate and joint continuity]]).
  
 
I. Namioka [[#References|[a10]]] proved that
 
I. Namioka [[#References|[a10]]] proved that
  
N) there is a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300106.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300107.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300108.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n1300109.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001010.png" />, the set of points of (joint) continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001011.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/n130010/n13001012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001013.png" />)]]). This is known as the [[Namioka theorem|Namioka theorem]].
+
N) there is a dense $G_\delta$-set $A$ contained in $X$ such that $A \times Y$ is contained in $C(f)$, the set of points of (joint) continuity of $f$ (cf. also [[Set of type F sigma(G delta)|Set of type $F_\sigma$ ($G_\delta$)]]). This is known as the [[Namioka theorem]].
  
Following [[#References|[a3]]], one says that a (Hausdorff) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001014.png" /> is a Namioka space if for any compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001015.png" />, any metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001016.png" /> and any separately continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001017.png" />, assertion N) holds.
+
Following [[#References|[a3]]], one says that a (Hausdorff) space $X$ is a Namioka space if for any compact space $Y$, any [[metric space]] $Z$ and any separately continuous function $f$, assertion N) holds.
  
J. Saint-Raymond [[#References|[a11]]] proved that separable Baire spaces are Namioka and all Tikhonov Namioka spaces are Baire; he also showed that in the class of metric spaces, Namioka and Baire spaces coincide (cf. also [[Baire space|Baire space]]).
+
J. Saint-Raymond [[#References|[a11]]] proved that [[Separable space|separable]] [[Baire space]]s are Namioka and all Namioka [[Tikhonov space]]s are Baire; he also showed that in the class of metric spaces, Namioka and Baire spaces coincide.
  
M. Talagrand [[#References|[a12]]] constructed an [[Alpha-favourable space|$\alpha$-favourable]] (hence, Baire) space that is not Namioka. It has been shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001022.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001023.png" />-defavourable spaces [[#References|[a11]]] and Baire spaces having dense subsets that are countable unions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001024.png" />-analytic subsets [[#References|[a5]]] are Namioka. The Sorgenfrey line is Namioka (cf. also [[Sorgenfrey topology|Sorgenfrey topology]]), although it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001025.png" />-favourable.
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M. Talagrand [[#References|[a12]]] constructed an [[Alpha-favourable space|$\alpha$-favourable]] (hence, Baire) space that is not Namioka. It has been shown that $\sigma$-$\beta$-defavourable spaces [[#References|[a11]]] and Baire spaces having dense subsets that are countable unions of $\mathbf{K}$-analytic subsets [[#References|[a5]]] are Namioka. The [[Sorgenfrey topology|Sorgenfrey line]] is Namioka , although it is $\alpha$-favourable.
  
Many permanence properties of Namioka spaces are known. In view of Saint Raymond's result, the Cartesian product of two (metric) Namioka spaces need not be Namioka. Also, Namioka spaces are not preserved, even in the metric case, by continuous perfect mappings (cf. also [[Blumberg theorem|Blumberg theorem]]).
+
Many permanence properties of Namioka spaces are known. In view of Saint Raymond's result, the Cartesian product of two (metric) Namioka spaces need not be Namioka. Also, Namioka spaces are not preserved, even in the metric case, by continuous [[perfect mapping]]s (cf. also [[Blumberg theorem]]).
  
Following G. Debs [[#References|[a5]]], one says that a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001026.png" /> is co-Namioka, or has the Namioka property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001027.png" /> (or belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001028.png" />) if for every Baire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001029.png" /> and for every semi-continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001030.png" />, the conclusion of Namioka's theorem holds. It was shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001031.png" /> holds for many compact-like spaces appearing in functional analysis; among them are Eberlein compact spaces [[#References|[a7]]], Corson compact spaces [[#References|[a6]]], Valdivia compact spaces [[#References|[a4]]], and, more generally, all compact spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001033.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001035.png" />-fragmentable [[#References|[a9]]]. It was shown by R. Deville [[#References|[a7]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001036.png" />. Recently (1999), A. Bouziad [[#References|[a1]]] showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001037.png" /> holds for all scattered compact spaces that are hereditarily submetacompact.
+
Following G. Debs [[#References|[a5]]], one says that a compact space $Y$ is co-Namioka, or has the Namioka property $N^*$ (or belongs to the class $\mathcal{N}^*$) if for every Baire space $X$ and for every semi-continuous function $f : X \times Y \rightarrow \mathbf{R}$, the conclusion of Namioka's theorem holds. It was shown that $N^*$ holds for many compact-like spaces appearing in functional analysis; among them are Eberlein compact spaces [[#References|[a7]]], Corson compact spaces [[#References|[a6]]], Valdivia compact spaces [[#References|[a4]]], and, more generally, all compact spaces $Y$ such that $C_p(Y)$ is $\sigma$-fragmentable [[#References|[a9]]]. It was shown by R. Deville [[#References|[a7]]] that $\beta N \not\in \mathcal{N}^*$. Recently (1999), A. Bouziad [[#References|[a1]]] showed that $N^*$ holds for all scattered compact spaces that are hereditarily submetacompact.
  
Certain permanence properties of co-Namioka spaces have been studied. For example, it is known that the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001038.png" /> is closed under continuous images, arbitrary products [[#References|[a2]]] and countable unions [[#References|[a8]]].
+
Certain permanence properties of co-Namioka spaces have been studied. For example, it is known that the class $\mathcal{N}^*$ is closed under continuous images, arbitrary products [[#References|[a2]]] and countable unions [[#References|[a8]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Bouziad,  "A quasi-closure preserving sum theorem about the Namioka property"  ''Topol. Appl.'' , '''81'''  (1997)  pp. 163–170</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bouziad,  "The class of co-Namioka compact spaces is stable under products"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 983–986</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  R. Deville,  G. Godefroy,  "Some applications of projective resolutions of identity"  ''Proc. London Math. Soc.'' , '''22'''  (1990)  pp. 261–268</TD></TR><TR><TD valign="top">[a5]</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><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Debs,  "Pointwise and uniform convergence on a Corson compact space"  ''Topol. Appl.'' , '''23'''  (1986)  pp. 299–303</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Deville,  "Convergence ponctuelle et uniforme sur un espace compact"  ''Bull. Acad. Polon. Sci.'' , '''37'''  (1989)  pp. 7–12</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Haydon,  "Countable unions of compact spaces with Namioka property"  ''Mathematika'' , '''41'''  (1994)  pp. 141–144</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.E. Jayne,  I. Namioka,  C.A. Rogers,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130010/n13001039.png" />-fragmentable Banach spaces"  ''Mathematika'' , '''41'''  (1992)  pp. 161–188; 197–215</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Namioka,  "Separate and joint continuity"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 515–531</TD></TR><TR><TD valign="top">[a11]</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">[a12]</TD> <TD valign="top">  M. Talagrand,  "Propriété de Baire et propriété de Namioka"  ''Math. Ann.'' , '''270'''  (1985)  pp. 159–174</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Bouziad,  "A quasi-closure preserving sum theorem about the Namioka property"  ''Topol. Appl.'' , '''81'''  (1997)  pp. 163–170</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bouziad,  "The class of co-Namioka compact spaces is stable under products"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 983–986</TD></TR>
 +
<TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  R. Deville,  G. Godefroy,  "Some applications of projective resolutions of identity"  ''Proc. London Math. Soc.'' , '''22'''  (1990)  pp. 261–268</TD></TR>
 +
<TR><TD valign="top">[a5]</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>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Debs,  "Pointwise and uniform convergence on a Corson compact space"  ''Topol. Appl.'' , '''23'''  (1986)  pp. 299–303</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Deville,  "Convergence ponctuelle et uniforme sur un espace compact"  ''Bull. Acad. Polon. Sci.'' , '''37'''  (1989)  pp. 7–12</TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Haydon,  "Countable unions of compact spaces with Namioka property"  ''Mathematika'' , '''41'''  (1994)  pp. 141–144</TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top">  J.E. Jayne,  I. Namioka,  C.A. Rogers,  "$\sigma$-fragmentable Banach spaces"  ''Mathematika'' , '''41'''  (1992)  pp. 161–188; 197–215</TD></TR>
 +
<TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Namioka,  "Separate and joint continuity"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 515–531</TD></TR>
 +
<TR><TD valign="top">[a11]</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">[a12]</TD> <TD valign="top">  M. Talagrand,  "Propriété de Baire et propriété de Namioka"  ''Math. Ann.'' , '''270'''  (1985)  pp. 159–174</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 21:29, 2 January 2016

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, let $Z$ be a pseudo-metric space, and let $f$ be an arbitrary separately continuous function (cf. also Separate and joint continuity).

I. Namioka [a10] proved that

N) there is a dense $G_\delta$-set $A$ contained in $X$ such that $A \times Y$ is contained in $C(f)$, the set of points of (joint) continuity of $f$ (cf. also Set of type $F_\sigma$ ($G_\delta$)). This is known as the Namioka theorem.

Following [a3], one says that a (Hausdorff) space $X$ is a Namioka space if for any compact space $Y$, any metric space $Z$ and any separately continuous function $f$, assertion N) holds.

J. Saint-Raymond [a11] proved that separable Baire spaces are Namioka and all Namioka Tikhonov spaces are Baire; he also showed that in the class of metric spaces, Namioka and Baire spaces coincide.

M. Talagrand [a12] constructed an $\alpha$-favourable (hence, Baire) space that is not Namioka. It has been shown that $\sigma$-$\beta$-defavourable spaces [a11] and Baire spaces having dense subsets that are countable unions of $\mathbf{K}$-analytic subsets [a5] are Namioka. The Sorgenfrey line is Namioka , although it is $\alpha$-favourable.

Many permanence properties of Namioka spaces are known. In view of Saint Raymond's result, the Cartesian product of two (metric) Namioka spaces need not be Namioka. Also, Namioka spaces are not preserved, even in the metric case, by continuous perfect mappings (cf. also Blumberg theorem).

Following G. Debs [a5], one says that a compact space $Y$ is co-Namioka, or has the Namioka property $N^*$ (or belongs to the class $\mathcal{N}^*$) if for every Baire space $X$ and for every semi-continuous function $f : X \times Y \rightarrow \mathbf{R}$, the conclusion of Namioka's theorem holds. It was shown that $N^*$ holds for many compact-like spaces appearing in functional analysis; among them are Eberlein compact spaces [a7], Corson compact spaces [a6], Valdivia compact spaces [a4], and, more generally, all compact spaces $Y$ such that $C_p(Y)$ is $\sigma$-fragmentable [a9]. It was shown by R. Deville [a7] that $\beta N \not\in \mathcal{N}^*$. Recently (1999), A. Bouziad [a1] showed that $N^*$ holds for all scattered compact spaces that are hereditarily submetacompact.

Certain permanence properties of co-Namioka spaces have been studied. For example, it is known that the class $\mathcal{N}^*$ is closed under continuous images, arbitrary products [a2] and countable unions [a8].

References

[a1] A. Bouziad, "A quasi-closure preserving sum theorem about the Namioka property" Topol. Appl. , 81 (1997) pp. 163–170
[a2] A. Bouziad, "The class of co-Namioka compact spaces is stable under products" Proc. Amer. Math. Soc. , 124 (1996) pp. 983–986
[a3] J.P.R. Christensen, "Joint continuity of separately continuous functions" Proc. Amer. Math. Soc. , 82 (1981) pp. 455–461
[a4] R. Deville, G. Godefroy, "Some applications of projective resolutions of identity" Proc. London Math. Soc. , 22 (1990) pp. 261–268
[a5] G. Debs, "Points de continuité d'une fonction séparément continue" Proc. Amer. Math. Soc. , 97 (1986) pp. 167–176
[a6] G. Debs, "Pointwise and uniform convergence on a Corson compact space" Topol. Appl. , 23 (1986) pp. 299–303
[a7] R. Deville, "Convergence ponctuelle et uniforme sur un espace compact" Bull. Acad. Polon. Sci. , 37 (1989) pp. 7–12
[a8] R. Haydon, "Countable unions of compact spaces with Namioka property" Mathematika , 41 (1994) pp. 141–144
[a9] J.E. Jayne, I. Namioka, C.A. Rogers, "$\sigma$-fragmentable Banach spaces" Mathematika , 41 (1992) pp. 161–188; 197–215
[a10] I. Namioka, "Separate and joint continuity" Pacific J. Math. , 51 (1974) pp. 515–531
[a11] J. Saint-Raymond, "Jeux topologiques et espaces de Namioka" Proc. Amer. Math. Soc. , 87 (1983) pp. 499–504
[a12] M. Talagrand, "Propriété de Baire et propriété de Namioka" Math. Ann. , 270 (1985) pp. 159–174
How to Cite This Entry:
Namioka space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Namioka_space&oldid=37299
This article was adapted from an original article by Z. Piotrowski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article