On the minimal cover property and certain notions of finite
type: Item
instance of: scholarly article
title: On the minimal cover property and certain notions of finite
zbMATH Open document ID: 1446.03083
author: Eleftherios Tachtsis
published in: Archive for Mathematical Logic
publication date: 2018-08-16T00:00:00Z
review text: Let \(X\) be a topological space. A cover \(\mathcal U\) of \(X\) is said to be a minimal cover, if for each \(U \in \mathcal U\), the family \(\mathcal U \setminus \{ U \}\) is not a cover of \(X\). \(X\) has the minimal cover property (mcp), if for each cover \(\mathcal U\) of \(X\) there exists a minimal subcover \(\mathcal V \subseteq \mathcal V\). \textsf{MCC} is the following principle: Every topological space with mcp is compact. In \textsf{ZFC}, \textsf{MCC} is true. But this is not the case in \textsf{ZF}. It was shown in [\textit{P. Howard} and \textit{E. Tachtsis}, Topology Appl. 173, 94--106 (2014; Zbl 1305.03038)] that a topological space with mcp is countably compact. In this paper, the author investigates the deductive strength of the principle ``every topological space with mcp is compact''. This is done with respect of notions of finite as well as with properties of linearly ordered sets and partially ordered sets. These properties concern the existence of countably infinite subsets for infinitely ordered sets, the existence of infinite chains or infinite antichains in infinite partially ordered sets, the existence of cofinal well-founded subsets of partially ordered sets. The author works with Fraenkel-Mostowski permutation models and Pincus' transfer theorems.
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 6919832
zbMATH Keywords: axiom of choice
zbMATH Keywords: weak axioms of choice
zbMATH Keywords: minimal cover property
zbMATH Keywords: compact space
zbMATH Keywords: notions of finite
zbMATH Keywords: partially ordered sets
zbMATH Keywords: linearly ordered sets
zbMATH Keywords: Fraenkel-Mostowski model
zbMATH Keywords: Pincus Transfer Theorem
MaRDI profile type: Publication
full work available at URL: s00153-017-0595-y
OpenAlex ID: W2769845943
cites work: Q5798522
cites work: Ramsey's theorem in the hierarchy of choice principles
cites work: Finiteness and choice
cites work: On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice
cites work: Well ordered subsets of linearly ordered sets
cites work: Definitions of finite
cites work: Q4210469
cites work: On the set-theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements
cites work: Axioms of multiple choice
cites work: The strength of the \(\Delta\)-system lemma
cites work: On the minimal cover property in \(\mathbf{ZF}\)
cites work: The axiom of choice
cites work: The independence of various definitions of finiteness
cites work: Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods
cites work: Adding dependent choice
cites work: Q4288131
cites work: Dependences between definitions of finiteness
cites work: On Ramsey's theorem and the existence of infinite chains or infinite anti-chains in infinite posets
cites work: Classes of Dedekind finite cardinals
cites work: The structure of amorphous sets
DOI: 10.1007/S00153-017-0595-Y
Recommended article: On the minimal cover property in \(\mathbf{ZF}\)
Recommended article: Q4855730
Recommended article: Minimal structures for \(T_{FA}\)
Recommended article: Ordinal compactness
Recommended article: Q4248088
Recommended article: Q5488198
Recommended article: The Minimal Structures for TA
Recommended article: Q3404188
Recommended article: On sequentially compact subspaces of \(\mathbb R\) without the axiom of choice
Recommended article: Galvin's question on non-\( \sigma \)-well ordered linear orders
description: scientific article; zbMATH DE number 6919832