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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on chains of open sets

Author: John Ginsburg
Journal: Proc. Amer. Math. Soc. 89 (1983), 317-325
MSC: Primary 54A25; Secondary 03E35
MathSciNet review: 712644
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Abstract: We consider some questions concerning the nature and size of chains of open sets in Hausdorff spaces. The following results are obtained.

Theorem 1. For every cardinal $ \kappa $ there exists a space $ X$ in which all discrete subsets have cardinality at most $ \kappa $ and which contains a chain of $ {({2^\kappa })^ + }$ open sets.

Theorem 2. If $ X$ is regular and contains a chain of $ {({2^\kappa })^ + }$ open sets, then $ X \times X$ contains a discrete subset of cardinality $ {\kappa ^ + }$.

Theorem 3. Let $ M(X)$ denote the set of all maximal chains of open subsets of $ X$ endowed with the Tychonoff topology. (i) $ \left\vert {M(X)} \right\vert \leqslant {2^{{\text{w}}(X)}}$, and (ii) $ \psi (M(X)) \leqslant {\text{w}}(X)$. Here $ {\text{w}}(X)$ denotes the weight of the space $ X$ and $ \psi (M(X))$ denotes the pseudocharacter of the space $ M(X)$.

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Keywords: Chains of open sets, discrete subset, partially ordered set, $ \kappa $-generated, maximal chains
Article copyright: © Copyright 1983 American Mathematical Society

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