# Countable spaces and countable dynamics

Perez Flores, Dolores Columba (2018). Countable spaces and countable dynamics. University of Birmingham. Ph.D.

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## Abstract

In this work we present results on three different topics. In the first of them, we consider the following situation: given a pair of functions $$f:X→Y$$ and $$g:X→Z$$, under which conditions can we find compact Hausdorff topologies on $$X$$,$$Y$$ and $$Z$$ with respect to which $$f$$ and $$g$$ are simultaneously continuous? We give a partial solution to the problem, solution that involves the One-point Compactification of a discrete space topology. Secondly, we extend the body of existing results on countable dynamical systems, which arise naturally in many dynamical settings. Among other results, we prove that these systems are ubiquitous in interval maps. The third part of this thesis is devoted to the study of the ordering by embeddability as a closed subset of closed sets of the real line. We characterise the poset $$2^R/\sim$$, where $$\sim$$ denotes the mentioned relation. The structure of countable compact Hausdorff spaces is the underlying notion that unifies this work.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Good, ChristopherUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: Other
Other Funders: Mexican Secretariat of Public Education, Consejo Nacional de Ciencia y Tecnología, Mexico
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/8105

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