Transfinite dimensions and mappings
type: Item
instance of: scholarly article
title: Transfinite dimensions and mappings
zbMATH Open document ID: 0653.54024
published in: Moscow University Mathematics Bulletin
publication date: 1987-01-01T00:00:00Z
review text: First, an estimate is derived for the transfinite inductive dimensions Ind X and ind X of a normal space X, in terms of the dimension of a subset \(A\subset Y\) (for ind X assume A closed in X) and the relative dimension of its complement. Let \(d=Ind\) or ind. Then \[ dX\leq rd_ X(X- A)+dA+1\quad if\quad dA<\omega_ 0 \] and \[ dX\leq rd_ X(X-A)+dA\quad if\quad dA\geq \omega_ 0. \] (Recall that \(rd_ XY=\sup \{dF|\) F closed in X and \(F\subset Y\}\), for \(Y\subset X\) an arbitrary subset of X.) This result is then applied to prove that certain maps, e.g. closed maps of bounded multiplicity or open maps of countable multiplicity, preserve the property of a normal space that it has transfinite dimension, provided that the range of the map is hereditarily normal.
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 4065910
zbMATH Keywords: transfinite inductive dimensions
zbMATH Keywords: normal space
zbMATH Keywords: relative dimension
zbMATH Keywords: closed maps of bounded multiplicity
zbMATH Keywords: open maps of countable multiplicity
author: V. V. Fedorchuk
reviewed by: Dušan Repovš
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4065910