Talk:Uniform space

message: There was a second definition that was basically a corrupt variation of the definition of a proximity space. Although there is a close relationship between proximity and uniformity, the two are distinct notions.
:I added the second definition thinking poximinity was the same concept as uniformity. The last few days I tried unsucessfully to prove this or find any reference. Thanks for fixing it. Can you point me to any reference on proximity space ? MathMartin 10:54, 24 Feb 2005 (UTC)
replies: text: I will be adding references to the proximity space page.--192.35.35.35 13:38, 24 Feb 2005 (UTC)

text: Yes you are correct, I meant to write quasiuniformity.MathMartin 10:54, 24 Feb 2005 (UTC)
replies: text: This is in line with "quasimetric" and "quasiproximity".--192.35.35.35 13:38, 24 Feb 2005 (UTC)

text: A one by one translation is probably not useful but generally there should be a note explaining the idea behind the definition.MathMartin 10:54, 24 Feb 2005 (UTC)
replies: text: OK, I'll be rewriting it. I wanted to be bold and just change it first, but seeing how long it has been part of the article, I didn't want to ruffle any feathers.--192.35.35.35 13:38, 24 Feb 2005 (UTC)

text: I added the reference, but it can be removed. MathMartin 10:54, 24 Feb 2005 (UTC)
replies: text: I'll be adding some more uniform space specific references.--192.35.35.35 13:38, 24 Feb 2005 (UTC)
thread_title:  Definition 

message: So which is the entourage, Φ or the elements of Φ?  The wording suggests that it's the elements of Φ, but then the definition of entourage imposes requirements on Φ, not just on its elements.  If the elements of Φ are called entourages, what is Φ itself called?  "Set of entourages" seems insufficiently specific. What if you have distinct Φ and Φ', each satisfying the axioms, and you take some elements of each to form a third set Φ3 that doesn't satisfy the axioms?  Isn't it also a set of entourages?  All its elements are entourages. --Trovatore 1 July 2005 01:31 (UTC)
::Found my answer in the topology glossary, and fixed the article (probably was just a typo) --Trovatore 1 July 2005 02:02 (UTC)
Something seems still funny about this definition. It stipulates a defining relation between Φ and U such that Φ is "a nonempty collection ... of subsets U...", but then goes on to repeat "if U \in Φ..." again and again as if the contrary could possibly be the case even though it was defined not to be. Is this just a disorganized mess or is there some implicit logic the reader is supposed to divine? Are there other objects U that are presumed to exist besides the ones that the label U was introduced to define?  — Preceding unsigned comment added by 24.56.246.116 (talk) 18:54, 9 September 2022 (UTC)
thread_title: Something fishy about the entourage definition

message: i fixed the given definition of entourages in completion space.
old: Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃A⊆F∩F*, A×A ⊆ U } be an entourage on Y.
new: Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∀ G≈F, G*≈F* ∃A∈G∩G*, A×A ⊆ U } be an entourage on Y
i think the definition of the entourages of the completion space using the round filter is OK (is given later on) but is NOT equivalent to the first definition.
first i suppose it meant:  Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃A∈F∩F*, A×A ⊆ U } be an entourage on Y. otherwise it makes no sense at all - but that's just type error.
this definition is wrong. for example in the real completion of rationals. let r be a real number, i take an increasing sequence of rationals converges to r and one decreasing converges to r-1. let F, F* be the smallest filters spanned by these two sequences, let U be the entourage of dist(x,y)<1, let A=[r-1,r]∩Q. then A∈F∩F* and A×A ⊆ U but it is not true that dist(r-1,r)<1. if i had chosen the smallest round filters then A∈F∩F* would be wrong (also for any other A st A×A ⊆ U).
i think it should be (which i put in there): Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∀ G≈F, G*≈F* ∃A∈G∩G*, A×A ⊆ U } be an entourage on Y
or the equivalent: Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃A∈G∩G* ∀ G≈F, G*≈F* , A×A ⊆ U } be an entourage on Y
use the properties of the unique round fiter to show equivalence of my two sugestions (it is the intersection of F/≈). --itaj 21:26, 19 August 2007 (UTC)
thread_title:  definition of entourages of completion space 

message: By http://eom.springer.de/U/u095250.htm metrizability as uniform space is not the same as metrizability as topological space: "In particular, a metrizable topology can be generated by a non-metrizable separating uniformity."
The current Wikipedia article contains the sentences "A Hausdorff uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff." so it complies with the definition of metrizable uniform space in the Springer EOM. However, the word "metrizable" refers to the article "Metrizable space" which talks about metrizable topological spaces, which is misleading.
So the articles Uniform space and Metrizable space should stress the distinction between metrizable uniform space and metrizable topological space (probably separate articles should be created for metrizable topological space and and metrizable uniform space).
--Jaan Vajakas (talk) 21:58, 16 May 2010 (UTC)
thread_title:  Metrizable uniform space 

message: The section "Uniformizable spaces" should refer to Uniformizable space. I think the proof that the topology of X is generated by C(X) should be moved to the article Uniformizable space as well. If I was not wrong in the previous comment that metrizability of uniform spaces is not the same as metrizability of topological spaces then the paragraph about metrizability should not belong to "Topology of uniform spaces" but a section of its own.--Jaan Vajakas (talk) 22:20, 16 May 2010 (UTC)
thread_title:  Uniformizable space 

message: The "uniform cover"- and "star refinement"-based definition of uniform spaces does not define the concept of uniform cover itself. Neither have I been able to find it defined elsewhere on Wikipedia. Somebody knowledgeable in this field please repair this.  — Preceding unsigned comment added by 109.100.71.77 (talk) 14:31, 24 February 2013 (UTC)
replies: text: A uniform cover is any cover which satisfies the axioms laid out in that section Forty-Bot (talk) 21:15, 1 March 2019 (UTC)
thread_title:  Definition of "uniform cover"