Abstract
Connectedness is an important property that a subset of a metric space may possess. Various propositions give ways of showing that a subset is connected. In, \(R\) the connected subsets are precisely the intervals. Connectedness is preserved by continuous functions, a fact that is the generalization of the Intermediate Value Theorem of real analysis. Every metric space decomposes uniquely as a union of connected disjoint components. Metric spaces can vary in their connectedness properties, from being connected, with a single component, to being totally disconnected, having single points as components
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© 2014 Springer International Publishing Switzerland
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Muscat, J. (2014). Connectedness. In: Functional Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-06728-5_5
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DOI: https://doi.org/10.1007/978-3-319-06728-5_5
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06727-8
Online ISBN: 978-3-319-06728-5
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