Baire property of some function spaces

• Published in: Acta mathematica Hungarica 2022-10, Vol.168 (1), p.246-259
• Main Authors: Osipov, A. V., Pytkeev, E. G.
• Format: Article
• Language: English
• Subjects: Continuity (mathematics); Function space; Mathematics; Mathematics and Statistics; Metric space; Topology
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-022-01274-7

A compact space X is called π - monolithic if for any surjective continuous mapping f : X → K where K is a metrizable compact space there exists a metrizable compact space T ⊆ X such that f ( T ) = K . A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X . Let C p ( X , Y ) denote the space of all continuous Y -valued functions C(X,Y) on a Tychonoff space X with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space X the space C p ( X , { 0 , 1 } ) is Baire if, and only if, C p ( X , K ) is Baire for every π -monolithic compact space K . For a Tychonoff space X the space C p ( X , R ) is Baire if, and only if, C p ( X , L ) is Baire for each Fréchet space L . We construct a totally disconnected Tychonoff space T such that C p ( T , M ) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, C p ( T , [ 0 , 1 ] ) is Baire but C p ( T , { 0 , 1 } ) is not.