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Set of continuity points of functions with values in generalized metric spaces. (English) Zbl 1212.54040

The paper considers the continuity points of functions with values in generalized metric spaces. For a function targeted to a weakly developable space, the notion of generalized oscillation is defined as a substitute for the oscillation (they can differ in metrizable spaces). A function with values in a weakly developable space is continuous at a point exactly when its generalized oscillation at this point is 0. The set of continuity points of such a function is \(G_\delta \). Another result obtained is that a topological space has a \(G_\delta \)-diagonal whenever it has the above property for any function targeted to it.
The set of continuity points of a quasicontinuous function from a Baire space to a weakly developable space is dense \(G_\delta \). This is used to prove that, if \(X\) is first countable, \(Y\) is Baire and \(Z\) is a regular weakly developable space, then, for any \(f\:X\times Y\to Z\) with quasicontinuous sections \(f_x\) and continuous sections \(f_y\), the set of continuity points of \(f\) lying in \(\{x\}\times Y\) is dense \(G_\delta \) therein thus extending a result for \(Z\) a Moore space.
The set of continuity points of a function with a closed graph to a \(p\)-space \(Y\) is shown to be \(G_\delta \). This remains true if \(Y\) is a \(w\Delta \)-space, provided that the domain space is first countable. If \(f\) is a continuous bijective map from a hereditarily Baire \(p\)-space, such that the points of continuity of \(f^{-1}\) form a dense set, then the target space contains a dense Baire subspace (so it is Baire itself).

MSC:

54C05 Continuous maps
54C08 Weak and generalized continuity
54E52 Baire category, Baire spaces
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