Namioka space

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