Continuing horrors of topology without choice

I. J. Tree; Christopher Good

doi:10.1016/0166-8641(95)90010-1

Continuing horrors of topology without choice

I. J. Tree^*, Christopher Good

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

38 Citations (Scopus)

Abstract

Various topological results are examined in models of Zermelo-Fraenkel set theory that do not satisfy the Axiom of Choice. In particular, it is shown that the proof of Urysohn's Metrization Theorem is entirely effective, whilst recalling that some choice is required for Urysohn's Lemma. R is paracompact and ω₁ may be paracompact but never metrizable. An example of a nonmetrizable paracompact manifold is given. Suslin lines, normality of LOTS and consequences of Countable Choice are also discussed.

Original language	English
Pages (from-to)	79-90
Number of pages	12
Journal	Topology and its Applications
Volume	63
Issue number	1
DOIs	https://doi.org/10.1016/0166-8641(95)90010-1
Publication status	Published - 21 Apr 1995

Keywords

Axiom of Choice
Suslin line
Urysohn's Lemma
Urysohn's Metrization Theorem
ω

ASJC Scopus subject areas

Geometry and Topology

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10.1016/0166-8641(95)90010-1

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AB - Various topological results are examined in models of Zermelo-Fraenkel set theory that do not satisfy the Axiom of Choice. In particular, it is shown that the proof of Urysohn's Metrization Theorem is entirely effective, whilst recalling that some choice is required for Urysohn's Lemma. R is paracompact and ω1 may be paracompact but never metrizable. An example of a nonmetrizable paracompact manifold is given. Suslin lines, normality of LOTS and consequences of Countable Choice are also discussed.

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KW - Suslin line

KW - Urysohn's Lemma

KW - Urysohn's Metrization Theorem

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Continuing horrors of topology without choice

Abstract

Keywords

ASJC Scopus subject areas

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