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A chaotic space

A chaotic space Acta Mathematiea Academiae Seientiarum Hungaricae Tomus 25 (1--2), (1974), pp. 1--4. By E. S. BERNEY (Pocatello) In this paper, assuming the Continuum Hypothesis, an example of a separable metric space is given which has cardinality e, which is linearly ordered (the linear order inducing the topology), and which has the following property; p, q EX such that p#q, implies there exist open sets Up (containing p) and U~ (containing q) such that no non-empty open subset of Up is homeomorphic to an open subset of Uq. A non-trivial topological space with this latter property is said to be chaotic. This result gives a positive answer to parts A and B of the following problem posed by E. D. Nix in [2, p. 957]: (A) Do chaotic spaces exist? (B) Do chaotic spaces of cardinality e exist? (C) Do there exist completely normal, connected, and locally connected chaotic spaces ? 1. Preliminaries. Let I denote the unit interval. Let A={(x, x):xCI}. If A is a set, let card A denote the cardinality of A. Let e=card I as well as the first ordinal of cardinality c. Let N denote the natural numbers. A subset A of a topological space will http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

A chaotic space

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01901738
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Abstract

Acta Mathematiea Academiae Seientiarum Hungaricae Tomus 25 (1--2), (1974), pp. 1--4. By E. S. BERNEY (Pocatello) In this paper, assuming the Continuum Hypothesis, an example of a separable metric space is given which has cardinality e, which is linearly ordered (the linear order inducing the topology), and which has the following property; p, q EX such that p#q, implies there exist open sets Up (containing p) and U~ (containing q) such that no non-empty open subset of Up is homeomorphic to an open subset of Uq. A non-trivial topological space with this latter property is said to be chaotic. This result gives a positive answer to parts A and B of the following problem posed by E. D. Nix in [2, p. 957]: (A) Do chaotic spaces exist? (B) Do chaotic spaces of cardinality e exist? (C) Do there exist completely normal, connected, and locally connected chaotic spaces ? 1. Preliminaries. Let I denote the unit interval. Let A={(x, x):xCI}. If A is a set, let card A denote the cardinality of A. Let e=card I as well as the first ordinal of cardinality c. Let N denote the natural numbers. A subset A of a topological space will

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Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Jun 18, 2005

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