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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.

285 questions
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What are interesting properties of totally bounded uniform spaces?

I work on reloids, a generalization of uniform spaces. As such I am interested about properties of totally bounded uniform spaces. My question: What are interesting properties of totally bounded uniform spaces? (The more theorems/propositions you…
porton
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A criterion of total boundness of a uniform space

Is the following true: Conjecture A uniform space $(U;F)$ is totally bounded iff for every entourage $E$ of this space there exists a finite set $B\subseteq U$ and a natural $n$ such that $E^n[B] = U$. If not, could you provide a counter-example?
porton
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Topology induced by uniformity (open and closed)

If V is an open (closed) in product topology X$\times$X that induced by uniformity , then $V(x) $ is open (closed) in $X$ ?
Ahmed Al Khuzai
  • 43
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Is $\mathcal B_Y=\{B\cap(Y\times Y):B\in\mathcal B\}$ forms a base for $\mathcal U_Y?$

In course of self-studying uniform space I have been stuck in some fundamental question: Let $(X,\mathcal U)$ be a uniform space and $Y\subset X.$ Then $\mathcal U_Y=\{U\cap(Y\times Y):U\in\mathcal U\}$ forms an uniformity on $Y$ to be called the…
Jave
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Base of Fine Uniformity

How to show that all normally open covers form a base for the fine uniformity $\mu_F$? If $\mathcal{B}$ is the collection of all normally open covers, we first need to show that $\mathcal{B}$ is a subcollection of $\mu_F$. Then we have to show that…
Mary Ku
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How to geometrically interpret the composition of entourages?

In my topology course, we have been working for quite some time now with uniform spaces. I understand the metric background and I can work abstractly with uniformities, so that is great. However, it has been bothering me since the beginning that I…
Jeroen
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Entourage definition to pseudometrics definition of uniform spaces

If $X$ is a set equipped with a collection $(d_i)_{i\in I}$ of pseudometrics, then the corresponding uniform structure (collection of entourages) $\Phi$ is defined by declaring that $U\in\Phi$ if and only if there exists $n\in\mathbb N$,…
zxcv
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Proof that a certain space equipped with preuniformity is a uniform space.

I'm using the uniform cover definition of a uniform space. Let $f:(X, \mu)\to (Y, \nu)$ be a surjective map such that $(X, \mu)$ is a uniform space and $\nu$ is a largest preuniformity on $Y$ such that if $\mathcal{U}\in\nu$ then…
Jakobian
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Uniform spaces, Dugundji chapter 9, section 11

I think I'm encountering a lot of errors in the section of Dugundji about uniform spaces. Let me write out my concerns. We now use a uniformity to derive a topology. Any uniformity $\mathscr{F}$ in $Y$ gives a topology $\mathscr{T}(\mathscr{F})$ in…
Jakobian
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Topology that induced by uniform structure

Let $(X,d)$ a metric space .Calculate the topology induced by uniformity that induced by pseudo metric d ( $U_{d} $)? Is topology induced by metric ? where $U_d=\{v\subseteq X\times X : v_\epsilon \subseteq v\} $ and $ v_\epsilon =\{(x,y)\in X…
Ahmed Al Khuzai
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On uniform Structure induced by pseudo metric

If $ U^{'} $ induced by pseudo metric d, then the induced uniform structure on $ X $ induced by the pseudo metric $ d(f\times f) $?
Ahmed Al Khuzai
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On induced Uniform Structure

Let $f:R \to (R,U^{'})$ be a mapping defined by: $f(x)$ =0 if $x\in Q$ and $f(x)$ = 1 if $x\in Q^{c}$. Find the uniform structure induced by $f$ , if $ (R,U^{'}) $ is uniform space induced by pseudo metric (discrete uniform space , indiscrete…
Ahmed Al Khuzai
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The restriction function is a uniformly continuous mapping

If A is a subset of a uniform space $ X $ and if $ f:X\to X^{'} $ is a uniformly continuous mapping , then the restriction $ f_{A}:A\to X^{'} $ is a uniformly continuous mapping of $ A $ into $ X^{'} $?
Ahmed Al Khuzai
  • 43
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Topology induced by Uniform Structure?

Let $f:X \to Y$ and $g:Y \to (Z,U^{''}) $ be mappings and $ U^{'} $ be induced uniform structures on $ Y $ by $ g $ and $ U $ be induced uniform structures on $ X $ by $ f $ , $ U_{1} $ be induced uniform structures on $ X $ by $ g \circ f $. Is…
Ahmed Al Khuzai
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Coarsest topology in Uniform Space

The topology on $X$ induced by the coarsest uniformity $U$ for which the the mappings $ f_i ,i\in I $ are uniformly continuous is also the coarsest topology for which the $ f_i ,i\in I $ , are continuous?
Ahmed Al Khuzai
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