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A metric characterization of connectedness for topological spaces

Weiss, Ittay (2014) A metric characterization of connectedness for topological spaces. [Professional and Technical Reports]

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Abstract

Connectedness, path connectedness, and uniform connectedness are well-known concepts. In the traditional presentation of these concepts there is a substantial difference between connectedness and the other two notions, namely connectedness is defined as the absence of disconnectedness, while path connectedness and uniform connectedness are defined in terms of connecting paths and connecting walks, respectively. In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. Connectedness in topological spaces can also be defined in terms of chains governed by open coverings in a manner that is more reminiscent of path connectedness. We present a metric formalism for connectedness which unifies all of the mentioned approaches to connectedness. The resulting connectedness criterion is valid for all topological spaces.

Item Type: Professional and Technical Reports
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Technology and Environment (FSTE) > School of Computing, Information and Mathematical Sciences
Depositing User: Ittay Weiss
Date Deposited: 20 Mar 2015 11:58
Last Modified: 12 Sep 2016 12:14
URI: http://repository.usp.ac.fj/id/eprint/8112
UNSPECIFIED

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