Namioka theorem

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].

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
[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