Summary
Open and closed subsets of a metric space are at the heart of this chapter. The concept of density has also been introduced. We have explained the notion of a metric topology. We have shown in detail how the topology of a metric subspace relates to that of an enveloping superspace and have determined the topology for conserving metrics on a finite product. We have demonstrated that ℝ with its usual metric is complete—in other words, universally closed—and have begun a related discussion about nests of closed subsets.
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© 2007 Springer-Verlag London Limited
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(2007). Open, Closed and Dense Subsets. In: Metric Spaces. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-84628-627-8_4
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DOI: https://doi.org/10.1007/978-1-84628-627-8_4
Publisher Name: Springer, London
Print ISBN: 978-1-84628-369-7
Online ISBN: 978-1-84628-627-8
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