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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Characterization of some zero-dimensional separable metric spaces


Author: Jan van Mill
Journal: Trans. Amer. Math. Soc. 264 (1981), 205-215
MSC: Primary 54F50; Secondary 54D99
DOI: https://doi.org/10.1090/S0002-9947-1981-0597877-9
MathSciNet review: 597877
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Abstract: Let $ X$ be a separable metric zero-dimensional space for which all nonempty clopen subsets are homeomorphic. We show that, up to homeomorphism, there is at most one space $ Y$ which can be written as an increasing union $ \cup _{n = 1}^\infty {F_n}$ of closed sets so that for all $ n \in {\mathbf{N}}$, $ {F_n}$ is a copy of $ X$ which is nowhere dense in $ {F_{n + 1}}$. If moreover $ X$ contains a closed nowhere dense copy of itself, then $ Y$ is homeomorphic to $ {\mathbf{Q}} \times X$ where $ {\mathbf{Q}}$ denotes the space of rational numbers. This gives us topological characterizations of spaces such as $ {\mathbf{Q}} \times {\mathbf{C}}$ and $ {\mathbf{Q}} \times {\mathbf{P}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0597877-9
Article copyright: © Copyright 1981 American Mathematical Society

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