Positional dimension-like functions of the type Ind
type: Item
instance of: scholarly article
title: Positional dimension-like functions of the type Ind
zbMATH Open document ID: 1230.54028
published in: Topology and its Applications
publication date: 2011-10-11T00:00:00Z
review text: In his master thesis [Subspace-dimension with respect to total spaces (1998)], \textit{R. Koga} gave a positional dimension-like function of the type Ind. Earlier papers of the authors and V. V. Tkachuk studied universalitity and dimensional properties of a positional dimension-like function of the type ind. In this paper it is shown that positional dimensions of type Ind can be stated in several manners. Let \(Q\) be a subset of \(X\). A family \({\mathcal B}\) of open subets of \(X\) is said to be a \(p(0)\)-big base (pos\((0)\)-big base) for \(Q\) in \(X\) if for every pair \((F,U)\) of subsets of \(X\), where \(F\) is closed in \(X\) and \(U\) is open in \(X\) and \(F\subset Q\cap U\), there exists a \(V\in{\mathcal B}\) such that \(F\subset Q\cap V\subset Q\cap U\) \((F\subset V\subset U)\). When we replace \(F\) closed in \(X\) by \(F^Q\) closed in \(Q\) in the above definition we obtain the definitions of a \(p(1)\)-big base (pos\((1)\)-big base) for \(Q\) in \(X\). We define the positional dimensions of type Ind as follows having ordinal values: \(p_0(0)\)-\(\text{Ind}(Q,X)= -1\) iff \(Q= X=\varnothing\), \(p_0(0)\)-\(\text{Ind}(Q,X)\leq\alpha\) iff there exists a \(p(0)\)-big base \({\mathcal B}\) for \(Q\) in \(X\) such that for every \(U\in{\mathcal B}\) we have \(p_0(0)\)-\(\text{Ind}(Q\cap \text{Bd}_X(U),\text{Bd}_X(U))<\alpha\). \(p_1(0)\)-\(\text{Ind}(Q,X)=-1\) iff \(Q=\varnothing\), \(p_1(0)\)-\(\text{Ind}(Q,X)\leq\alpha\) iff there exists a \(p(0)\)-big base \({\mathcal B}\) for \(Q\) in \(X\) such that for every \(U\in{\mathcal B}\) we have \(p_1(0)\)-\(\text{Ind}(Q\cap\text{Bd}_X(U), X)<\alpha\). \(\text{pos}_0(0)\)-\(\text{Ind}(Q,X)\), \(\text{pos}_1(0)\)-\(\text{Ind}(Q,X)\) are defined by using pos\((0)\)-big bases in the above definitions. Using \(p(1)\)-big bases and pos\((1)\)-big bases we again define four dimension functions, \(p_0(1)\)-\(\text{Ind}(Q,X)\), \(p_1(1)\)-\(\text{Ind}(Q,X)\), \(\text{pos}_0(1)\)-\(\text{Ind}(Q,X)\) and \(\text{…
reviewed by: Piet Borst
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54B99
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 5956310
zbMATH Keywords: positional dimension-like function
zbMATH Keywords: large inductive dimension
author: Dimitris Georgiou
author: S. D. Iliadis
author: Athanasios Megaritis
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.045
OpenAlex ID: W2029789911
cites work: Q4855460
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DOI: 10.1016/J.TOPOL.2011.06.045
description: scientific article; zbMATH DE number 5956310