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Continuity in separable metrizable and Lindelöf spaces

Authors: Chris Good and Sina Greenwood
Journal: Proc. Amer. Math. Soc. 138 (2010), 577-591
MSC (2000): Primary 37B99, 54A10, 54B99, 54C05, 54D20, 54D65, 54H20; Secondary 37-XX
Published electronically: October 14, 2009
MathSciNet review: 2557175
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Abstract: Given a map $ T:X\to X$ on a set $ X$ we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on $ X$ with respect to which $ T$ is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is such a topology precisely when the cardinality of $ X$ is no greater than $ \mathfrak{c}$, the cardinality of the continuum. We go on to prove that there is a Lindelöf topology on $ X$ with respect to which $ T$ is continuous if either $ T^{\mathfrak{c}^+}(X)=T^{\mathfrak{c}^++1}(X)\neq\emptyset$ or $ T^\alpha(X)=\emptyset$ for some $ \alpha<\mathfrak{c}^+$, where $ T^{\alpha+1}(X)=T\big(T^\alpha(X)\big)$ and $ T^\lambda(X)=\bigcap_{\alpha<\lambda}T^{\alpha}(X)$ for any ordinal $ \alpha$ and limit ordinal $ \lambda$.

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Additional Information

Chris Good
Affiliation: School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom

Sina Greenwood
Affiliation: University of Auckland, Private Bag 92019, Auckland, New Zealand

Keywords: Abstract dynamical system, topological dynamical system, Lindel\"of, separable metric, hereditarily Lindel\"of
Received by editor(s): October 1, 2008
Published electronically: October 14, 2009
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.