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 B1(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 B1(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. P-espacios y un teorema incondicional de gráfica cerrada Resumen. Sea X un espacio completamente regular (Tychonoff). Por C(X), U (X) y B1(X) se denotan los conjuntos de funciones reales definidas en X que son continuas, que tienen gráfica cerrada y que son de primera clase de Baire, respectivamente. Se prueba que U (X) = C(X) si y sólo si X es un P espacio (es decir que cada subconjunot G δ de X es abierto) o si y sólo si B1(X) = U (X). Estos resultados extienden una relación de equivalencias obtenidas por Gillman y Henriksen, Onuchic e Iséki. La primera equivalencia es un teorema incondicional de gráfica cerrada; implica que si X es perfectamente normal o cumple el primer axioma de numerabilidad (por ejemplo si es metrizable), o es localmente compacto, entonces existen funciones discontinuas en X con gráfica cerrada. Así se complementan resultados obtenidos por Dobos y Bags sobre discontinuidad de funciones con gráfica cerrada.

Maximal classes for families of lower and upper semicontinuous functions with a closed graph

“…Proof The implication (⇐) we can find in [2] (see p. 118, lines [11][12][13][14]. The implication (⇒) immediately follows from [6]…”

“…It is easy to see that Remark 1 Let X be a topological space such that Finally, observe that we can extend the lists (see e.g. [11,Theorem 1]) of equivalent conditions for X to be a P-space as follows: Corollary 2.16 Let X be a nonempty completely regular space. Then X is a P-space if and only if M min (Ulsc(X )) = ∅.…”

On structure space of the ring B1(X)

“…In general, B 1 (X) = C(X) does not always imply that X is discrete. For example, if X is a P-space, then B 1 (X) = C(X) ( [7]). We shall now show that for a particular class of topological spaces, e.g., for perfectly normal T 1 spaces, B 1 (X) = C(X) is equivalent to the discreteness of the space.…”