2020/01/16 by S. Garcı́a-Ferreira, S. Garcia-Ferreira, Garcia-Ferreira, S. +2
Mathematics · #54C15 #54D30 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #Combinatorics #Compact space #Computer science #Discrete mathematics #Embedding #FOS: Mathematics #General Topology (math.GN) #Geometry #Linear subspace #Mathematical analysis #Mathematics #Monotone polygon #Pure mathematics #Separable space #Skeleton (computer programming) #Space (punctuation) #math.GN #msc:54C15 #msc:54D30
paper · pdf · doi:10.48550/arxiv.2001.06312
arXiv admin note: text overlap with arXiv:1804.01549
arxiv created 2020/01/16 · openalex publication_date 2020/01/16 · arxiv updated 2020/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
It is know that the Valdivia compact spaces can be characterized by a special family of retractions called r-skeleton (see \citekubis1). Also we know that there are compact spaces with r-skeletons which are not Valdivia. In this paper, we shall study r-squeletons and special families of closed subsets of compact spaces. We prove that if X is a zero-dimensional compact space and \rs:s∈ Γ\ is an r-skeleton on X such that |rs(X)| ≤ ω for all s∈ Γ, then X has a dense subset consisting of isolated points. Also we give conditions to an r-skeleton in order that this r-skeleton can be extended to an r-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a Σ-product of a power of the unit interval. Following this fact we introduce the notion of π-skeleton on a compact space X by embedding X in a suitable power of the unit interval together with a pair (F,φ), where F is family of metric separable subspaces of X and φ an ω-monotone function which satisfy certain properties. This new notion generalize the idea of a Σ-product. We prove that a compact space admits a retractional-skeleton iff it admits a π-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with retractional-skeletons admits an retractional-skeleton (see \citecuth1). In \citecasa1, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak c-skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with r-skeletons.