Questions tagged [uniform-spaces]

In the mathematical field of Topology, a uniform space is a generalization of the concept of metric spaces in which, unlike what happens in general topological spaces, it is possible to compare neighborhoods of distinct points. In uniform spaces, it is possible to define the concepts of uniform continuity, uniform convergence, and of complete space (as in the metric spaces, but unlike what happens in topological spaces in general).

In the mathematical field of topology, a uniform space is a set with a uniform structure.

Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence.

278 questions
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a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it immediate consequences to the general theory. For…
Ittay Weiss
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do the uniformly continuous functions to the reals determine the uniformity?

It is well known that the completely regular spaces $X$ are characterized as those topological spaces whose topology is recovered from $C(X)$, the set of continuous functions $X\to \mathbb R$. In other words, $C(X)$ determines the topology on $X$.…
Ittay Weiss
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What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are continuous. $d_1$ and $d_2$ are uniformly…
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Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) $\quad$ Can it be abelian? $\quad$ Can it be…
user57159
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Is $\operatorname{Homeo}([0,1])$ Weil-Complete?

After learning about uniformities on topological groups, we were given several sources to read. I came across the term "Weil-complete." A topological group is Weil-complete if it is complete with respect to the left (or right) uniformity. We learned…
josh
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Topological groups are completely regular

I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all completely regular, I wasn't able to either find a…
Patrick Da Silva
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Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It seems pretty staightforward, but I am having…
josh
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When is a uniform space complete

From Wikipedia: a uniform space is called complete if every Cauchy filter converges. I was wondering if the following three are equivalent in a uniform space: every Cauchy filter converges, every Cauchy net converges, and every Cauchy sequence…
Tim
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non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is there a similar such characterization of compactness…
Ittay Weiss
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Do $\Bbb H$ and $\Bbb R^2$ have the same uniform structure?

The hyperbolic upper half plane $\Bbb H$ and euclidean space $\Bbb R^2$ are not isomorphic as metric spaces, which can be see from the fact that in $\Bbb R^2$ for any point not on a geodesic there exists exactly one geodesic going through that point…
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Is a quotient of a complete group always complete?

Let $\: \langle G,\cdot,\mathcal{T}\hspace{0.01 in} \rangle \:$ be a $\big($$\text{T}_0$$\big)$ topological group. $\;\;$ Let $H$ be a closed normal subgroup of $G$. Set $\;\; \mathbf{G} \: = \: \langle G,\cdot,\mathcal{T}\hspace{0.01 in} \rangle…
user57159
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Separation axioms in uniform spaces

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ such that $U \circ V \subset W$, then $U[x]…
Bilbo
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Explicit description of a left adjoint to the forgetful functor from Unif to Top

Let $\newcommand\Unif{\mathrm{Unif}}\Unif$ denote the category of uniform spaces and uniformly continuous functions and $\newcommand\Top{\mathrm{Top}}\Top$ the category of topological spaces and continuous functions. We have a forgetful functor…
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non-hausdorff completion of a uniform space.

Let $(X,\mathcal U)$ be a Hausdorff uniform space. Can $(X,\mathcal U)$ have a non-hausdorff completion?
user79193
7
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Is a set bounded in every metric for a uniformity bounded in the uniformity?

This is a follow-up to my question here. A subset $A$ of a uniform space is said to be bounded if for each entourage $V$, $A$ is a subset of $V^n[F]$ for some natural number $n$ and some finite set $F$. A subset of a metric space is said to be…
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