Integer-valued functions and increasing unions of first countable spaces
type: Item
instance of: scholarly article
title: Integer-valued functions and increasing unions of first countable spaces
zbMATH Open document ID: 0711.54002
DOI: 10.1007/BF02937294
author: Alan Dow
author: István Juhász
author: William A. R. Weiss
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let \(\kappa\) be an infinite cardinal number, with \(\{X_{\alpha}:\alpha <\kappa \}\) a \(\kappa\)-chain of first countable spaces (i.e., \(X_{\alpha}\subseteq X_{\beta}\) for \(\alpha <\beta <\kappa)\). Let \(X=\cup_{\alpha <\kappa}X_{\alpha}\). Then there is a coarsest topology and a finest topology on X such that each \(X_{\alpha}\) is a subspace. Called the weak and fine topologies respectively in this paper, the question is raised as to when there is some compatible first countable topology on X. The following summarizes several interesting results the authors have obtained investigating this question. 1. If \(\kappa =\aleph_ 0\), then the weak topology is first countable and the fine topology is ``almost never'' first countable. 2. If \(\kappa\) is uncountable regular and X has a compatible first countable topology, then that topology must be the fine topology. 3. As for the main question, the one that asks for each uncountable regular \(\kappa\) whether the fine topology is first countable, the answer is absolutely ``no'' for \(\kappa =\aleph_ 1\), but is otherwise axiom- sensitive: The answer is ``yes'' if \(\kappa\) is either weakly compact or \(\geq\) some supercompact cardinal; if \(V=L\), then the answer is ``no'' unless \(\kappa\) is weakly compact. On the other hand, if there exists a supercompact cardinal, then forcing models can be constructed in which the answer is ``yes'' for every \(\kappa >\aleph_ 1.\) Several of the proofs exploit a translation of the main question into a mathematically equivalent question involving certain families of integer- valued functions.
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 4169625
zbMATH Keywords: first countable spaces
zbMATH Keywords: weak and fine topologies
zbMATH Keywords: integer-valued functions
reviewed by: Paul Bankston
MaRDI profile type: Publication
cites work: Increasing strengthenings of cardinal function inequalities
cites work: Set theory. An introduction to independence proofs
cites work: On the role of supercompact and extendible cardinals in logic
cites work: Aronszajn trees and the independence of the transfer property
full work available at URL: https://doi.org/10.1007/bf02937294
OpenAlex ID: W2053132109
description: scientific article; zbMATH DE number 4169625