The uniform-spaces tag has no wiki summary.

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### Utility representation on uniform spaces

I have been looking into conditions under which a binary relation can be represented by a utility function. Most papers in know of consider restrictions on the space ( almost always a ordered ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 =\{ ...

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### 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 ...

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### 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 ...

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### 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)| < ...

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### 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 ...

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### 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< ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\mathscr{R}$. ...

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### 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 ...

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### 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 $|\cdot|_i$, ...

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### 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 ...

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### 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 $n$ ...