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Is the term "complete topological space" used in some notable references? Otherwise I think it should be deleted from the article. Anyway, I think "complete topological space" should not used on Wikipedia, as it is misleading (explained below).
I think the term "complete topological space" is outright confusing: there exist complete topological vector spaces which are not metrizable (consider e. g. the space of all smooth functions on with compact supports, used as a test function space in distribution theory --- this is an LF-space which is complete but not metrizable (as explained in Distribution space#Test function space)), so they are not "complete topological spaces" in the sense of this article (although they are complete and they are topological spaces!). Actually I like the notion "topologically complete" only a little more, because it would be as natural to use it for complete uniformizability (like Kelley does), but it is used by several notable authors.
So I would rather delete the phrase "complete topological space" from this article (and any redirects and links to it), or if some notable authors have used "complete topological space" in this sense, then it should be mentioned in this article but it should be pointed out that this usage is misleading. -- Jaan Vajakas ( talk) 01:37, 5 March 2013 (UTC)
I am not happy with the sentences "Every metrizable space has a completion that is a completely metrizable space. The completion of Q is R." in the article. I guess metrizable spaces and metric spaces have been confused here. How would the completion of a metrizable topological space be defined? There are many choices for a complete metric space that is a completion of some metric on the given metrizable topological space. Maybe there indeed exists a natural choice of choosing a single one out of them, but I couldn't find such in Willard's book. E. g. the completion of (0, 1) with respect to one metric compatible with its topology is [0, 1], with respect to another one (0, 1) itself is complete, and these completions [0, 1] and (0, 1) are not topologically the same (I should come up with a better counterexample, since one could of course define the completion of a metrizable space to be the space itself in the special case when it is already completely metrizable). Jaan Vajakas ( talk) 10:24, 19 December 2013 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
Is the term "complete topological space" used in some notable references? Otherwise I think it should be deleted from the article. Anyway, I think "complete topological space" should not used on Wikipedia, as it is misleading (explained below).
I think the term "complete topological space" is outright confusing: there exist complete topological vector spaces which are not metrizable (consider e. g. the space of all smooth functions on with compact supports, used as a test function space in distribution theory --- this is an LF-space which is complete but not metrizable (as explained in Distribution space#Test function space)), so they are not "complete topological spaces" in the sense of this article (although they are complete and they are topological spaces!). Actually I like the notion "topologically complete" only a little more, because it would be as natural to use it for complete uniformizability (like Kelley does), but it is used by several notable authors.
So I would rather delete the phrase "complete topological space" from this article (and any redirects and links to it), or if some notable authors have used "complete topological space" in this sense, then it should be mentioned in this article but it should be pointed out that this usage is misleading. -- Jaan Vajakas ( talk) 01:37, 5 March 2013 (UTC)
I am not happy with the sentences "Every metrizable space has a completion that is a completely metrizable space. The completion of Q is R." in the article. I guess metrizable spaces and metric spaces have been confused here. How would the completion of a metrizable topological space be defined? There are many choices for a complete metric space that is a completion of some metric on the given metrizable topological space. Maybe there indeed exists a natural choice of choosing a single one out of them, but I couldn't find such in Willard's book. E. g. the completion of (0, 1) with respect to one metric compatible with its topology is [0, 1], with respect to another one (0, 1) itself is complete, and these completions [0, 1] and (0, 1) are not topologically the same (I should come up with a better counterexample, since one could of course define the completion of a metrizable space to be the space itself in the special case when it is already completely metrizable). Jaan Vajakas ( talk) 10:24, 19 December 2013 (UTC)