Families of retractions and families of closed subsets on compact spaces
arXiv:2001.06312
Abstract
It is know that the Valdivia compact spaces can be characterized by a special family of retractions called -skeleton (see \cite{kubis1}). Also we know that there are compact spaces with -skeletons which are not Valdivia. In this paper, we shall study -squeletons and special families of closed subsets of compact spaces. We prove that if is a zero-dimensional compact space and is an -skeleton on such that for all , then has a dense subset consisting of isolated points. Also we give conditions to an -skeleton in order that this -skeleton can be extended to an -skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a -product of a power of the unit interval. Following this fact we introduce the notion of -skeleton on a compact space by embedding in a suitable power of the unit interval together with a pair , where is family of metric separable subspaces of and an -monotone function which satisfy certain properties. This new notion generalize the idea of a -product. We prove that a compact space admits a retractional-skeleton iff it admits a -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 \cite{cuth1}). In \cite{casa1}, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak -skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with -skeletons.
arXiv admin note: text overlap with arXiv:1804.01549