Separate vs. Joint Continuity; A Tale of Four Topologies I
by
R.G. Woods
University of Manitoba
Coauthors: R. G. Woods (University of Manitoba)

Throughout, X, Y denote Tychonoff spaces, R the real line with its usual topology, C(X) all continuous f : X --> R, C*(X) the bounded elements of C(X), and S(X ×Y) the set of all separately continuous real-valued functions with domain X ×Y. Define sp : R² --> R by sp(x, y) = \frac2xyx²+ y² if (x, y) =/= (0, 0) and sp(0, 0) = 0. Clearly, sp in S(R²) \C(R²). Let H = C* (X ×Y) \cup { sp o (f ×g) : f in C(X) }

Four topologies are considered on the product X ×Y of two Tychonoff spaces: \tau is the usual product topology, \sigma is the weak topology generated by all the separately continuous real-valued functions, \lambda_H is the weak topology generated by H. For x in X and y in Y, call {x} ×Y a vertical section and X ×{y} a horizontal section. Define

\gamma = { U subset X ×Y : U \cap ( {x} ×Y) in \tau| {x} ×Y and U \cap (X ×{y}) in \tau| X ×{y} for all x in X, y in Y }.

Then \tau subset \lambda_H subset \sigma subset \gamma.

Theorem. C(X ×Y, \gamma) = C(X ×Y, \sigma) = S(X ×Y) and the compact subsets of (X ×Y, \lambda_H) are finite unions of compact subspaces of vertical or horizontal sections.

Theorem. The complete regularization of (X ×Y, \gamma) is (X ×Y, \sigma) and the k-space coreflection of (X ×Y, \sigma) is (kX ×kY, \gamma).

Theorem. If X is first countable, Y is a Baire space, then \tau\{\emptyset} is a \pi-base for \sigma.

Theorem. If X and Y are complete separable metric spaces with no isolated points, then \sigma =/= \gamma; if X = Y, then the diagonal of (X², \sigma) is discrete and so (X ×Y, \sigma) is not normal.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-96.