2022/04/12 by Alexander V. Osipov, Osipov, Alexander V., Е. Г. Пыткеев +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2204.05974
openalex publication_date 2022/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Cp(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 Cp(X,\0,1\) is Baire if, and only if, Cp(X,K) is Baire for every π-monolithic compact space K. For a Tychonoff space X the space Cp(X) is Baire if, and only if, Cp(X,L) is Baire for each Frechet space L. We construct a totally disconnected Tychonoff space T such that Cp(T,M) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, Cp(T,[0,1]) is Baire but Cp(T,\0,1\) is not.