So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R. For example, to show that any countable subspace of a metric space is disconnected, I had to come up with a distance d[itex]\in[/itex]R such that d isn't equal to any distances in the countable space. So I assume the completion of R must come before a metric may even be defined, which is a mapping into R; the definition of an open ball is even more dependent, since it is a bijective mapping from (X, R) to the set of all open balls in X. Does this sound about right?(adsbygoogle = window.adsbygoogle || []).push({});

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# Metric and completeness of real numbers

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