# Talk:Uniform space

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Field: Topology

## untitled

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)
I will be adding references to the proximity space page.--192.35.35.35 13:38, 24 Feb 2005 (UTC)

The term for something satisfying the first four entourage axioms is "quasiuniformity".--192.35.35.34 15:25, 23 Feb 2005 (UTC)

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

The Notes section, with the translation of the axioms one by one, is silly. Is there a reader who actually finds it helpful to be walked through all five translations? The result, to me, is far more confusing.--192.35.35.34 16:47, 23 Feb 2005 (UTC)

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)
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)

Why is Steen & Seebach in the references?--192.35.35.34 16:57, 23 Feb 2005 (UTC)

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

OK, I think my changes have been for the better. I plan to add material on pseudometric families, total boundedness, Cauchy completeness.--192.35.35.34 16:27, 24 Feb 2005 (UTC)

## Something fishy about the entourage definition

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)

## definition of entourages of completion space

i fixed the given definition of entourages in completion space.

old: Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃AFF*, A×AU } be an entourage on Y.

new: Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∀ GF, G*≈F* ∃AGG*, A×AU } 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*/≈ ) : ∃AFF*, A×AU } 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 AFF* and A×AU but it is not true that dist(r-1,r)<1. if i had chosen the smallest round filters then AFF* would be wrong (also for any other A st A×AU).

i think it should be (which i put in there): Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∀ GF, G*≈F* ∃AGG*, A×AU } be an entourage on Y

or the equivalent: Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃AGG* ∀ GF, G*≈F* , A×AU } 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)

The stub on Gauge spaces is an unnecessary separate article on one of the equivalent definitions of a uniform space. Moreover, the name does not appear to be much used. In any case, the minimal contents should be moved into this article under the third equivalent definition of uniforms space by means of pseudometrics. This subsection should also be expanded — now the pseudometric definition is given nowhere. Stca74 (talk) 11:42, 30 December 2007 (UTC)

Added material to Uniform space on the definition by means of pseudometrics. A further problem with the existing stub on Gauge spaces: it is not clear whether it is intended to mean a topological space the topology of which can be defined in terms of pseudometrics (in which case it is exactly the same as a uniformizable space or indeed a completely regular space) or a uniform space with a specific family of pseudometrics defining its uniformity. The latter would make more sense, which is why the proposed redirect is into the subsection on the pseudometrics definition of uniform structure. Stca74 (talk) 07:02, 2 January 2008 (UTC)
I totally agree with the merge. Cazort (talk) 02:22, 23 January 2008 (UTC)

Maybe this merger should be undone! Actually Gauge spaces (correct definition see for example [nLab entry]) are different from uniform spaces. Every gauge space defines a uniform structure and every uniform structure comes from a gauge space, that's true. Nevertheless gauge spaces have more structure: In a gauge space one can talk about bornological features (i.e. bounded sets and everything derived from that). This is also a feature of metric spaces that is lost during the metric-uniform transition, because every metric space is uniformly homeomorphic to a bounded metric space (e.g. $d'(x,y) = \min\{d(x,y),1\}$ or $d''(x,y):=\frac{d(x,y)}{1+d(x,y)}$.)

One way to clarify the distinction: The obvious functor {gauge spaces} --> {uniform spaces} has a left adjoint (assigning to each uniformity the *maximal set* of semimetrics inducing this uniformity) such that the composition {unif.} --> {gauge} --> {unif} is the identity. In this sense uniform spaces are gauge space with a special property (the set of semimetrics is saturated in a certain sense) and a uniform space is completely determined by its gauge structure. On the other hand: The composition {gauge} --> {unif} --> {gauge} is not the identity, that is: Gauge spaces are not determined by their uniform structure. 141.35.40.141 (talk) 16:10, 3 February 2015 (UTC)

## Metrizable uniform space

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)

## Uniformizable space

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)

## Definition of "uniform cover"

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)