Separate and joint continuity

It follows from a property of the product topology that every continuous function $f:X\times Y\to Z$ between topological spaces is separately continuous, i.e., $f$ is continuous with respect to each variable while the other variable is fixed [a3]. It was observed by E. Heine [a14], p. 15, that, in general, the converse does not hold (see also [a11] for an account of early discoveries in this field).

Given "nice" topological spaces $X$ and $Y$, let $M$ be a metric space and let $f:X\times Y\to M$ be separately continuous. Questions on separate and joint continuity are, among others, problems of the type:

the existence problem: Find the set $C(f)$ of points of continuity of $f$.

If $X$ and $Y$ are "nice" , then $C(f)$ is a dense $G_\delta$-subset (cf. Set of type $F_\sigma$ ($G_\delta$)) of $X\times Y$. For example, every real-valued separately continuous function $f:\textbf{R}^2\to\textbf{R}$ is of the first Baire class (cf. also Baire classes), hence $C(f)$ is a dense $G_\delta$-subset.

There is also interest in a "fibre" version; it is the same as above, except now one looks for $C(f)$ in $\{x\}\times Y$, for any fixed $x\in X$.

The characterization problem. Characterize $C(f)$ as a subset of $X\times Y$

If $X=Y=M=\textbf{R}$, then the set $C(f)$ is the complement of an $F_\sigma$-set contained in the product of two sets of the first Baire category [a8].

The uniformization problem. Find out whether there is a dense $G_\delta$-subset $A$ of $X$ such that the set $A\times Y$ is contained in $C(f)$.

If $X=Y=M=\textbf{R}$, such a result was known already to R. Baire [a1]. If $X$ is a complete metric space, $Y$ is a compact metric space and $M=\textbf{R}$, the uniformization problem was positively solved by H. Hahn [a7]. I. Namioka [a9] extended Hahn's result to $X$ being a regular, strongly countably complete topological space, $Y$ being a locally compact $\sigma$-compact space and $Z$ being a pseudo-metric space.

Following [a4], one says that $X$ is a Namioka space if for any compact space $Y$, any metric space $M$ and any separately continuous function $f$, the uniformization problem can be positively solved. M. Talagrand [a13] constructed an $\alpha$-favourable space (hence, a Baire space) that is not Namioka. J. Saint Ramond [a12] proved that separable Baire spaces are Namioka, Namioka Tikhonov spaces are Baire, while in the class of metric spaces, the set of Namioka spaces and the set of Baire spaces coincide.

A space $Y$ is a co-Namioka space if for any Baire space $X$, any metric space $M$ and for any separably continuous function $f$, the uniformization problem can be positively solved. For example, Corson-compact spaces are co–Namioka, whereas the Čech–Stone compactification $\beta N$ is not. Many results in this direction have been obtained by R. Haydon, R.W. Hansell, J.E. Jain, J.P. Troallic, Namioka, and R. Pol (see [a10] for a comprehensive exposition of this topic, organizing research in this field until the middle of the 1980s).

Applications.

R. Ellis [a5], [a6] showed that every locally compact semi-topological group (i.e., a group endowed with a topology for which the product is separately continuous) is a topological group. Using methods of separate and joint continuity, A. Bouziad [a2] extended Ellis' theorem to all Čech-analytic Baire semi-topological groups (hence, to all Čech-complete semi-topological groups).

References

[a1]	R. Baire, "Sur les fonctions des variables réelles" Ann. Mat. Pura Appl. , 3 (1899) pp. 1–122
[a2]	A. Bouziad, "Every Čech-analytic Baire semitopological group is a topological group" Proc. Amer. Math. Soc. , 124 (1996) pp. 953–959
[a3]	A.L. Cauchy, "Cours d'Analyse de l'Ecole Polytechnique" (1821)
[a4]	J.P.R. Christensen, "Joint continuity of separately continuous functions" Proc. Amer. Math. Soc. , 82 (1981) pp. 455–462
[a5]	R. Ellis, "A note on the continuity of the inverse" Proc. Amer. Math. Soc. , 8 (1957) pp. 372–373
[a6]	R. Ellis, "Locally compact transformation groups" Duke Math. J. , 24 (1957) pp. 119–125
[a7]	H. Hahn, "Reelle Funktionen" , Akad. Verlag (1932) pp. 325–338
[a8]	R. Kershner, "The continuity of functions of many variables" Trans. Amer. Math. Soc. , 53 (1943) pp. 83–100
[a9]	I. Namioka, "Separate and joint continuity" Pacific J. Math. , 51 (1974) pp. 515–531
[a10]	Z. Piotrowski, "Separate and joint continuity" Real Anal. Exch. , 11 (1985-86) pp. 293–322
[a11]	Z. Piotrowski, "The genesis of separate versus joint continuity" Tatra Mtn. Math. Publ. , 8 (1996) pp. 113–126
[a12]	J. Saint Raymond, "Jeux topologiques et espaces de Namioka" Proc. Amer. Math. Soc. , 87 (1983) pp. 499–504
[a13]	M. Talagrand, "Espaces de Baire et espaces de Namioka" Math. Ann. , 270 (1985) pp. 159–164
[a14]	J. Thomae, "Abriss einer Theorie der complexen Funktionen" , Louis Nebert Verlag (1873) (Edition: Second)