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2
votes
1answer
78 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 ...
1
vote
1answer
74 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 \...
5
votes
3answers
113 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) = ...
1
vote
0answers
42 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 ...
5
votes
0answers
116 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 ...
6
votes
2answers
348 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 ...
4
votes
1answer
137 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 ...
4
votes
1answer
217 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 ...
2
votes
1answer
118 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 $A\cap B=\{0\}$?
7
votes
0answers
134 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 ...
4
votes
0answers
151 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 ...
3
votes
1answer
572 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 ...
4
votes
2answers
142 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 ...
1
vote
1answer
90 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^...
0
votes
2answers
180 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 ...
1
vote
1answer
106 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,...
2
votes
1answer
255 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
1answer
145 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 ...
1
vote
1answer
79 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)| < \...
3
votes
3answers
320 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 ...
0
votes
2answers
171 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_{...
2
votes
1answer
361 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
votes
1answer
466 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 ...
3
votes
2answers
198 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 ...
2
votes
1answer
250 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 ...
3
votes
1answer
226 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 ...
12
votes
2answers
430 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 $\mathscr{R}$. (...
3
votes
3answers
387 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 ...
4
votes
1answer
237 views

Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
8
votes
4answers
1k views

Finite dimensional vector spaces over a complete but not-necessarily-valued field

I'm essentially reopening this old question of Ricky Demer which was never fully answered. Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
3
votes
1answer
329 views

Chaos in uniform spaces

Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying: For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural number $n$ ...