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P-spaces and an unconditional closed graph theorem

P-espacios y un teorema incondicional de gráfica cerrada

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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.

Resumen

Sea X un espacio completamente regular (Tychonoff). Por C(X), U(X) y B 1(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 P-space (i.e., every G δ de X es abierto) o si y sólo si B 1(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.

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Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85-072, Bydgoszcz, Poland

    Marek Wójtowicz & Waldemar Sieg

Authors
  1. Marek Wójtowicz
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  2. Waldemar Sieg
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Correspondence to Marek Wójtowicz.

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Submitted by Manuel Valdivia

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Wójtowicz, M., Sieg, W. P-spaces and an unconditional closed graph theorem. Rev. R. Acad. Cien. Serie A. Mat. 104, 13–18 (2010). https://doi.org/10.5052/RACSAM.2010.03

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  • DOI: https://doi.org/10.5052/RACSAM.2010.03

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