Between compactness and completeness |
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Authors: | Gerald Beer |
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Affiliation: | Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA 90032, USA |
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Abstract: | Call a sequence in a metric space cofinally Cauchy if for each positive ε there exists a cofinal (rather than residual) set of indices whose corresponding terms are ε-close. We give a number of new characterizations of metric spaces for which each cofinally Cauchy sequence has a cluster point. For example, a space has such a metric if and only each continuous function defined on it is uniformly locally bounded. A number of results exploit a measure of local compactness functional that we introduce. We conclude with a short proof of Romaguera's Theorem: a metrizable space admits such a metric if and only if its set of points having a compact neighborhood has compact complement. |
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Keywords: | primary, 54E50 secondary, 54E45 |
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