P-espacios y un teorema incondicional de gráfica cerrada

Abstract

Let X be a completely regular (Tychonoff) space, and let C(X), U(X), and B 1(X) denote the sets of all real-valued functions on X that are continuous, have a closed graph, and of the first Baire class, respectively. We prove that U(X) = C(X) if and only if X is a P-space (i.e., every G δ -subset of X is open) if and only if B 1(X = U(X). This extends a list of equivalences obtained earlier by Gillman and Henriksen, Onuchic, and Iséki. The first equivalence can be regarded as an unconditional closed graph theorem; it implies that if X is perfectly normal or first countable (e.g., metrizable), or locally compact, then there exist discontinuous functions on X with a closed graph. This complements earlier results by Doboš and Baggs on discontinuity of closed graph functions.