Questions tagged [uniform-spaces]
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65 questions
6 votes
2 answers
721 views
Approximate a continuous function by uniformly continuous ones
Let $(X, d)$ be a complete and separable metric space. I am interested in the case where bounded subsets of $X$ are not necessarily compact. Let $f: X \to \mathbb R$ be bounded and continuous. Is ...
- 1,163
4 votes
0 answers
226 views
Does every compact metric space admit a finite contracting family of maps?
Let $X$ be a metric space. A collection $\mathcal F$ of maps $X\to X$ (not necessarily continuous) is called contracting if $$\forall \varepsilon>0\ \exists n\in\mathbb N\ \forall f_1,\dots,f_n\in\...
- 1,100
0 votes
0 answers
103 views
Existence of maps between two modules over almost algebras living in an entourage. Lemma 2.3.7 in Gabber, Ramero
I would like to ask about the proof of Lemma 2.3.7 in "Almost Ring Theory" by Gabber and Ramero. Their proof proves part (iv) of the claim and I am specifically having trouble with the line: ...
- 9
4 votes
3 answers
412 views
Non metrizable uniform spaces
Bourbaki's book on general topology states that a uniform space is metrizable iff it is Hausdorff and the filter of entourages of the uniformity has a countable basis. However, he doesn't provide an ...
- 225
3 votes
1 answer
154 views
Even covers and collectionwise normal spaces
We call $X$ strongly collectionwise normal if the set $\mathcal{U}_\Delta$ of all neighbourhoods of the diagonal $\Delta_X$ of $X\times X$ is a uniformity. This is equivalent to the property that for ...
- 2,287
3 votes
1 answer
272 views
Extensions of bounded uniformly continuous functions
Let $X$ be a uniform space, $S\subseteq X$ and $f:S\to \mathbb{R}$ bounded uniformly continuous, then there exists a uniformly continuous extension of $f$ to $X$. (Katětov, 1951) I am looking for ...
- 2,287
3 votes
0 answers
113 views
Categorical characterization of Hausdorff completion of a uniform space
I know that a function between uniform spaces $\varphi:X\to X'$ is an Hausdorff completion if and only if: $X'$ is complete Hausdorff uniform space; $\varphi$ is a dense and initial uniformly ...
- 289
3 votes
0 answers
133 views
What is final uniformity in general?
On any set $X$ a (not necessarily set-indexed) family of functions $(f_i\colon X\to Y_i)_{i\in I}$ to uniform spaces $(Y_i)_{i\in I}$ induces a coarsest uniformity which makes all $f_i$ uniformly ...
- 131
3 votes
0 answers
193 views
What is the universal/fine uniformity on a topological group?
Cross posted from https://math.stackexchange.com/questions/4889335 I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\...
- 1,522
1 vote
1 answer
141 views
Image of a complete topological group under open and surjective map is complete?
A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff. Question. Let $G$ be a complete topological group and let $H$ be a topological ...
- 542
7 votes
3 answers
1k views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 341
7 votes
0 answers
421 views
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 struck me that the definition of Grothendieck Topology bears some familiar ...
- 803
1 vote
0 answers
65 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{...
1 vote
1 answer
239 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 ...
- 1,643
2 votes
1 answer
191 views
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 ...
- 1,643
4 votes
0 answers
168 views
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 ...
- 4,616
12 votes
0 answers
264 views
Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and 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 ...
- 2,805
13 votes
2 answers
2k 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\...
- 3,338
0 votes
1 answer
121 views
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 ...
- 1
9 votes
6 answers
1k 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 ...
3 votes
1 answer
318 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 ...
- 44k
2 votes
1 answer
178 views
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 ...
- 19k
39 votes
3 answers
3k views
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:(...
- 4,573
4 votes
1 answer
211 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\...
- 141
4 votes
1 answer
993 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 ...
- 195
3 votes
1 answer
664 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 ...
- 31
8 votes
1 answer
578 views
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 ...
- 261
4 votes
1 answer
204 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}...
- 63
0 votes
1 answer
113 views
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?
- 195
4 votes
1 answer
299 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 ...
- 195
7 votes
0 answers
287 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 ...
- 10.4k
8 votes
0 answers
333 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 ...
- 44k
2 votes
1 answer
130 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 ...
- 2,169
3 votes
1 answer
167 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 the ...
- 153
2 votes
1 answer
151 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 \...
- 153
5 votes
3 answers
231 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) = ...
- 166
1 vote
0 answers
117 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 ...
- 11
5 votes
0 answers
160 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 ...
- 1,165
14 votes
2 answers
2k 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 ...
- 1,280
4 votes
1 answer
408 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 ...
- 851
4 votes
1 answer
374 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 ...
- 165
3 votes
1 answer
222 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\}$?
- 1,165
11 votes
1 answer
347 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 ...
- 251
5 votes
0 answers
373 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 ...
- 5,477
6 votes
1 answer
3k 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 ...
user5810
5 votes
2 answers
257 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 ...
1 vote
1 answer
122 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^n=...
user35674
0 votes
2 answers
250 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)$$ A ...
user31967
1 vote
1 answer
205 views
Normal Uniform Spaces and their function uniform spaces
Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase $$\Lambda =\{ \{(f,...
user31967
2 votes
1 answer
436 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 $\...
