Separate versus joint continuity: A tale of four topologies

Trivial and simple spectrum for SL(d, ℝ) cocycles with free base and fiber dynamics

“…In what follows, we endow the product space Aut Leb (M ) × L ∞ (M, SL(d, R)) with the separate topology in W ×L ∞ . For full details on the issue of continuity of multivariable real maps, see [11].…”

Semicoverings: a generalization of covering space theory

“…This topology is sometimes called the cross topology [16]. Let X ⊗ Y be the set-theoretic product with the weak topology with respect to the projection fibers …”

Baire and weakly Namioka spaces

“…Answering Problem C, p. 203 of [11], the first-named author [22] refines Brown's techniques ("two-dimensional" example) by constructing a separately continuous real-valued function f defined on the Cartesian product of two complete metric spaces X, Y such that the (in fact, dense G δ ) set C(f ) of points of (joint) continuity fails to contain either A × Y or X × B for any dense G δ -set A in X or any dense G δ set B in Y . In other words, the condition of the theorem for Namioka spaces fails "in both directions".…”