Questions tagged [uniform-spaces]

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Generalizing uniform structures as Grothendieck topologies

Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
Nik Pronko's user avatar
1 vote
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33 views

extension from a dense subset in completely uniformizable spaces

Let $\mathbf{CReg}$ the category of completely regular spaces and continuous maps and let $\mathbf{Unif}$ be the category of uniform spaces and uniformly continuous maps. There is a functor $F:\mathbf{...
Ruben Van Belle's user avatar
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1 answer
181 views

A question about uniformities generated by pseudometrics

Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X $. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left( a_{n}\right) $ is a sequence of positive ...
Mehmet Onat's user avatar
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1 answer
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A a question about the metrization of uniform spaces

I have read two theorems about the metrization of uniform spaces from Engelking and Kelley. Kelley's condition (b) is slightly different from Engelking's corresponding result for Vi's. I think these ...
Mehmet Onat's user avatar
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4 votes
0 answers
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Alternative uniformities on topological groups

Are there any interesting alternative uniformities defined on topological groups besides the usual four (left, right, and their meet/join)? I am curious because in the (sort of) dual setting of coarse ...
Cameron Zwarich's user avatar
12 votes
0 answers
195 views

Do compact inverse-property loops (or just compact Moufang loops) have bi-invariant Haar measure?

So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
Harry Altman's user avatar
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13 votes
2 answers
1k views

Uniform spaces as condensed sets

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Unif{Unif}\DeclareMathOperator\CHaus{CHaus}\DeclareMathOperator\Set{Set}\DeclareMathOperator\op{op}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\...
Z. M's user avatar
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What is the definition of a prorelation?

In the context of quasi-uniform spaces, what is a prorelation? In the text I'm reading, they're defined as a down-directed upper set on relations X->Y. Now, I'm fine with a down-directed up-set, but ...
4amvim's user avatar
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9 votes
6 answers
732 views

Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
J. van Dobben de Bruyn's user avatar
4 votes
1 answer
229 views

Does each $\omega$-narrow topological group have countable discrete cellularity?

A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable. A family $\mathcal F$ of subsets of a topological space ...
Taras Banakh's user avatar
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2 votes
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Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question. Let $X$ be a set with a uniform structure ...
James Hanson's user avatar
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36 votes
3 answers
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What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
Keshav Srinivasan's user avatar
4 votes
1 answer
152 views

What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

It is known that if $(X, d)$ is a compact metric space, then hyperspace $2^X= \{A\subseteq X: A\text{ is closed set} \}$ is a compact space with Hausdorff metric What can say about $2^X= \{A\...
user479859's user avatar
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1 answer
721 views

Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper1. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have ...
Jave's user avatar
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3 votes
1 answer
540 views

Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...
VMrcel's user avatar
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1 answer
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Totally bounded spaces and axiom of choice

Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...
C. Eratosthene's user avatar
4 votes
1 answer
158 views

When is the unitary dual of a lscs group uniformizable?

Let $G$ be a locally compact, second countable group. We equip the unitary dual $\widehat{G}$ with the Fell topology. I am looking for conditions which guarantee that the topological space $\widehat{G}...
Tom's user avatar
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1 answer
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Topology generated by complete and incomplete uniformities [closed]

Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?
Jave's user avatar
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1 answer
234 views

Convergent net in a quasi-uniform space which is not Cauchy

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...
Jave's user avatar
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197 views

Results that are easier in a metric space

Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces? In particular, I'm ...
Alec Rhea's user avatar
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278 views

Has the Roelcke completion of a topological group any reasonable algebraic structure?

It is well-known that each topological group $G$ carries (at least) four natural uniformities: the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...
Taras Banakh's user avatar
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2 votes
1 answer
99 views

Cartesian powers of uniform spaces

In the nlab entry on uniform spaces they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped ...
geodude's user avatar
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3 votes
1 answer
120 views

Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. To construct Fine uniformities, Let ...
M. A.'s user avatar
  • 153
2 votes
1 answer
111 views

The separated uniform space associated with $(X,\mathfrak{U})$

If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \...
M. A.'s user avatar
  • 153
5 votes
3 answers
174 views

A Mackey-Ahrens theorem for uniform spaces?

Let $X$ be a uniform space and $F(X)$ the vector space of all uniformly continuous real-valued functions over $X$. It is possible to express every bounded uniform semimetric $d$ on $X$ as $d(x,y) = ...
Matthias's user avatar
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1 vote
0 answers
97 views

In a topological group $G$ with its lower uniformity, if $G$ is locally totally bounded, is its completion locally compact?

There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ...
PIH's user avatar
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5 votes
0 answers
136 views

Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
Minimus Heximus's user avatar
13 votes
2 answers
1k views

Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
Jonathan Gleason's user avatar
4 votes
1 answer
294 views

In the category of uniform spaces, is the completion of a quotient map also a quotient map?

I asked this question about 2 months ago on math.stackexchange, but so far I received neither comments nor answers. Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous ...
J.-E. Pin's user avatar
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4 votes
1 answer
313 views

The subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) : Let $(X,\mathcal{U})$ be a uniform ...
Jay's user avatar
  • 165
3 votes
1 answer
144 views

Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
Minimus Heximus's user avatar
11 votes
1 answer
280 views

Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
Jamie Walton's user avatar
5 votes
0 answers
307 views

The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ...
Alex M.'s user avatar
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4 votes
1 answer
2k views

How is the notion of a Lipschitz structure on a manifold defined?

According to wikipedia, there is such a definition. $\:$ The candidate that I can come up with is "an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz to ...
user avatar
5 votes
2 answers
223 views

A theorem of Markov about completely regular spaces and topological groups

In Pontriaguin's classic book Grupos continuos (in English Continuous Groups), says that A. Markov proved that: There are topological groups that are not normal. Furthermore, he says it is deduced ...
Josué Tonelli-Cueto's user avatar
1 vote
1 answer
99 views

Existence of a moderate uniform structure on $\Bbb R$

A moderate uniform structure $\mathcal U$ on $\Bbb R$ is one for which $\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$ but $ \not\exists n\in \Bbb N,\forall U\in \mathcal U,\quad U^...
user avatar
0 votes
2 answers
207 views

Is there a normal space that is not uniformly normal

Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which $$(\forall a\in A)(D[a]\subseteq B)$$ A ...
user avatar
1 vote
1 answer
155 views

Normal Uniform Spaces and their function uniform spaces

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase $$\Lambda =\{ \{(f,...
user avatar
2 votes
1 answer
393 views

Extend Homeomorphism to Uniformly Continuous Function

I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$. I'm trying to build a CW-complex with it, so I want a continuous function from the closed ball $\overline{B}_n$ to the closure $\...
David Collins's user avatar
1 vote
1 answer
162 views

Precompact reflection in diagonal uniform spaces

Each diagonal uniform space $(X,\mathcal D)$ can be derived from the covering uniform space $(X,\Sigma_{\mathcal D})$ and each covering uniform space $(X,\Sigma)$ can be derived from the diagonal ...
user avatar
1 vote
1 answer
128 views

Reference: uniformity of pointwise convergence has no countable base

Does anyone have a reference for the fact that the uniformity of pointwise convergence on real functions of $[0,1]$ (that is, the uniformity generated by the sets $\lbrace (f,g) : |f(x) - g(x)| < \...
Andrew Poelstra's user avatar
3 votes
3 answers
403 views

Complete uniform spaces require complete metrics?

Hey all, It is well-known that any uniformity is generated by the family of pseudometrics which are uniformly continuous from the product uniformity to $\mathbb{R}$. Further, the uniformity is ...
Andrew Poelstra's user avatar
2 votes
2 answers
299 views

Uniformities generated by metrics.

Any uniformity on a set $X$ is generated by a family of pseudometrics on $X$. So if $(X,\mathcal D)$ is a uniform space there's a set $P$ of pseudometrics on $X$ with $$\mathcal D=\left< \bigcup_{...
user avatar
1 vote
1 answer
649 views

Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces

This is a follow up question to this one. If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm). Do we have $...
AlexE's user avatar
  • 2,896
2 votes
1 answer
1k views

Extending uniformly continuous functions on subspaces to non-metrizable compactifications

I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$. Furthermore, I do have a uniformly continuous function $f$ on $X$. So ...
AlexE's user avatar
  • 2,896
3 votes
2 answers
223 views

For any entourage $U,V$ there's an entourage $W$ such that $U\circ W\subseteq V\circ U$

Let $(X,\mathcal U)$ be a uniform space and let $U\in \mathcal U$. Is this statement true? $$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$ I think if the above ...
user31968's user avatar
2 votes
1 answer
606 views

A uniformity with a countable base is a pseudometric uniformity.

I need a proof for this proposition: If a uniformity $\mathfrak U$ on $X$ has a countable fundamental system of entourages, then it can be defined by a pseudometric on $X$. which is the ...
user31813's user avatar
2 votes
1 answer
235 views

Does every proximal outer measure, measure all open sets?

Let $\: \langle X,\delta\rangle \: $ be a separated proximity space. Let $\: \mu^* \: : \: 2^{X} \: \to \: [0,+\infty] \: $ be a proximal outer measure. Let $U$ be an open subset of $X$. Does it ...
user avatar
12 votes
2 answers
671 views

Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]

And what else can be said, if so? (Original math.SE post) In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (...
Harry Altman's user avatar
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4 votes
3 answers
499 views

Does every Lindelof uniform space have a Lindelof completion?

Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces? Note it is well known to be true for ...
Fred Dashiell's user avatar