Abstract
It is an open problem if any separable compact space K whose function space C(K) with the cylindrical σ-algebra is a standard measurable space, embeds in the space of the first Baire class functions on the Cantor set, with the pointwise topology. We prove that this is true for separable linearly ordered compacta.
Other problems discussed in this note concern the Borel structures in C(K) generated by the norm, weak or pointwise topology in C(K). We give an example of a compact space K such that the weak and the pointwise topology generate different Borel structures in C(K).
Resumen
Es un problema abierto saber si cada espacio compacto separable K cuyo espacio de funciones C(K) con la σ-álgebra cilíndrica es un espacio medible estándar, se sumerge en el espacio de las funciones de la primera clase de Baire definidas en el conjunto de Cantor, con la topología de convergencia puntual. Aquí probamos que lo anterior es cierto para compactos separables linealmente ordenados. Discutimos en esta nota otros problemas relativos a las estructuras de Borel en C(K) generadas por la norma, topología débil o topología de convergencia puntual en C(K). Damos un ejemplo de un espacio compacto K tal que la topología débil y la topología de convergencia puntual generan estructuras de Borel diferentes en C(K).
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Dedicated to Professor Manuel Valdivia on the occasion of his 80th birthday
Submitted by Fernando Bombal
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Marciszewski, W., Pol, R. On some problems concerning Borel structures in function spaces. RACSAM 104, 327–335 (2010). https://doi.org/10.5052/RACSAM.2010.20
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DOI: https://doi.org/10.5052/RACSAM.2010.20