Questions tagged [uniform-spaces]
The uniform-spaces tag has no usage guidance.
54
questions
6
votes
0
answers
204
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 stroke me that, that the definition of Grothendieck Topology bears some ...
- 447
1
vote
0
answers
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{...
0
votes
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 ...
- 1,109
1
vote
1
answer
96
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 ...
- 1,109
4
votes
0
answers
103
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 ...
- 3,222
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 ...
- 2,495
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\...
- 1,723
0
votes
1
answer
93
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
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 ...
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 ...
- 40.2k
2
votes
1
answer
128
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 ...
- 10.1k
36
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:(...
- 4,489
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\...
- 141
4
votes
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 ...
- 195
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 ...
- 31
7
votes
1
answer
478
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 ...
- 251
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}...
- 63
0
votes
1
answer
77
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
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 ...
- 195
7
votes
0
answers
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 ...
- 8,878
6
votes
0
answers
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 ...
- 40.2k
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 ...
- 2,099
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 ...
- 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 \...
- 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) = ...
- 166
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 ...
- 11
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 ...
- 1,145
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 ...
- 1,220
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 ...
- 811
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 ...
- 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 $A\cap B=\{0\}$?
- 1,145
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 ...
- 151
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 ...
- 5,105
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 ...
user5810
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 ...
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^...
user35674
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 ...
user31967
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
$$\Lambda =\{ \{(f,...
user31967
2
votes
1
answer
393
views
Extend Homeomorphism to Uniformly Continuous Function
I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$.
I'm trying to build a CW-complex with it, so
I want a continuous function from the closed ball $\overline{B}_n$
to the closure $\...
1
vote
1
answer
162
views
Precompact reflection in diagonal uniform spaces
Each diagonal uniform space $(X,\mathcal D)$ can be derived from the covering uniform space $(X,\Sigma_{\mathcal D})$ and each covering uniform space $(X,\Sigma)$ can be derived from the diagonal ...
user31967
1
vote
1
answer
128
views
Reference: uniformity of pointwise convergence has no countable base
Does anyone have a reference for the fact that the uniformity of pointwise convergence on real functions of $[0,1]$ (that is, the uniformity generated by the sets $\lbrace (f,g) : |f(x) - g(x)| < \...
- 245
3
votes
3
answers
403
views
Complete uniform spaces require complete metrics?
Hey all,
It is well-known that any uniformity is generated by the family of pseudometrics which are uniformly continuous from the product uniformity to $\mathbb{R}$. Further, the uniformity is ...
- 245
2
votes
2
answers
299
views
Uniformities generated by metrics.
Any uniformity on a set $X$ is generated by a family of pseudometrics on $X$. So if $(X,\mathcal D)$ is a uniform space there's a set $P$ of pseudometrics on $X$ with
$$\mathcal D=\left< \bigcup_{...
user31967
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 $...
- 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 ...
- 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 ...
- 63
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 ...
- 31
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 ...
user5810
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 $\mathscr{R}$. (...
- 2,495
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 ...
- 1,654