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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).

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Covering characterization of Metrizability and Paracompactness

I am quite struck by the similarity between covering charachterization of Paracompactness and Metrizability both in the T1 and T3 topologies. In particular in the case of T1 topologies we have the ...
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$\Delta_X$ is $\mu$- isolated set

Let $g:X\times X\to X\times X$ be a continuous function and $(X, \mathcal{U})$ be a compact uniform space. Also for entourage $D\in\mathcal{U}$ suppose \begin{equation} \{z\in X\times X: g^n(z)\in D,...
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Proof verification: totally bounded iff every filter has a cauchy superfilter

It would be appreciated if someone could check if these proofs are correct (in Schechter's HAF, the 'only if' direction of the theorem below is essentially left as an exercise with a hint, so I wanted ...
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Question Regarding Proposition 8.1.18 in Engelking

Proposition 8.1.18 in Engelking is stated without proof, and I am having trouble proving the second part of the assertion. It would be greatly appreciated if I received a hint. Note that Engelking ...
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Quasi-uniform spaces are not regular in general; where does this argument fail?

It is well known that every topological space is quasi-uniformizable, and every quasi-uniform space $(X, \mathcal U)$ induces a natural topology on $X$. Moreover, if $(X, \mathcal U)$ is a quasi-...
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Question about neighborhood of diagonal of a topological space

Let $X$ be a topological space but it is not uniform space and $U$ be a neighborhood of diagonal $X$, $\Delta_X$. Is there a neighborhood $D(\neq \Delta_X)$ of $\Delta_X$ such that $D\circ D \...
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Why coarse maps have to be proper?

A map $f: X \to Y$ between metric spaces is said to be coarse, if the following two conditions hold: $f$ is bornologous, i.e. $$\forall_{R>0} \; \exists_{S>0} \; d(x,y) < R \Rightarrow d(f(...
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Uniquely Promoting a TVS over the Rationals to a TVS over the Reals

I have a Topological Vector Space (TVS) ($I, \oplus , \otimes $) over $ \mathbb Q$ and I want to uniquely extend its scalar multiplication to $ \mathbb R $, so that it is promoted to a TVS over $ \...
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Every uncountable compact uniform space has a nonatomic measure?

It is known that the class of nonatomic measures on compact metric space without isolated point $X$ is dense in $\mathcal{M}(X)$, where $\mathcal{M}(X)$ is the set of Borel probability measures ...
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Question on the Proof to the Pseudometrization Lemma for (Quasi) Uniform Spaces

I am reading the proof to the following theorem from Cobzas "Functional Analysis in Asymmetric Normed Spaces" (this also appears in Kelley's General Topology), and there is a step of the proof I am ...
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Why are uniform spaces important?

Why are uniform spaces important? I've thought of two possible good answers: Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and ...
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Homeomorphic spaces are uniformly isomorphic

A continuous function $f$ is a homeomorphism if it is bijective, and open. A uniformly continuous function $f$ is a uniform isomorphism if it is bijective and $f^{-1}$ is uniformly continuous. Is ...
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Uniform Isomorphism

I have read somewhere that a function $f:(X,\mathcal{D}(X))\longrightarrow (Y,\mathcal{D}(Y))$ is a uniform isomorphism provided that $f$ and $f^{-1}$ are uniform continuous functions and $f$ is ...
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Let $N$ be neighborhood of diagonal of $X$ $\Delta_X$. Is it true $int(N)\neq \emptyset$? [closed]

Let $X$ be a topological space and $N$ be a neighborhood of the diagonal $X$. Let $\{x_n\}$ and $\{y_n\}$ be in $X$ with $(x_n, y_n)\notin N$. If $x_n\to x$ and $y_n\to y$, is it true that $x\neq y$ ?...
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Neighbourhood filter of a uniform space

The Wikipedia page for Uniform space includes the following section: Every uniform space $V$ becomes a topological space by defining a subset $O$ of $X$ to be open if and only if for every $x$ in $...
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Hausdorff completion of uniform spaces

I'm working on constructing a completion of uniform spaces. As far as possible I'd like to follow the construction in the case of metric spaces that uses equivalence classes of Cauchy sequences ...
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Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are ...
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Relation Between Nbhd Base at $e$ and the Uniform Structure on a Topological Group

We have the following theorem (from Husain's Introduction to Topologcal Groups), slightly rephrased: The following are true (apologies for the sloppy formatting): (1) in any topological group $G$, ...
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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 ...
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Is there a proper set in a locally compact, paracompact uniform space?

Let $X$ be a uniform space with uniformity $\mathcal{U}$ and natural topology of it is locally compact and paracompact space. (the natural topology, $\tau_{\mathcal{U}}$, on $X$ is the family of all ...
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Is there an entorage $N\subseteq \{(x, y): d(x, y)<\delta\}$ in an uniformity of locally compact metric space $(X, d)$?

It is known that if $\mathcal{U}$ be a uniformity on $X$, then every entourage $N\in\mathcal{U}$ is an open set of diagonal $\Delta_X$, but the converse of it, is not true. For instance, Consider $...
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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 ...
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totally disconnected compact Hausdorff uniform space

Let $(X, \mathcal{U})$ be a compact and Hausdorff uniform space. For $D\in\mathcal{U}$, the sequence $\{x_i\}_{i\in\mathbb{Z}}$ is called a $D$- chain if $(x_i, x_{i+1})\in D$ for all $i\in\mathbb{Z}$...
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What is the purpose of this axiom in the definition of a uniform space?

I started reading about uniform spaces the other day, and in an attempt to familiarize myself with the notion I attempted to work through some examples based on the entourage definition of a ...
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Is every uniformizable topology induced by a Heine-Borel 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 $...
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Neighbourhoods in a quasi-distance induced topology

A quasi-distance on a set $X$ is a non-negative function $\delta(x,y)$ defined on $X\times X$ such that: $\delta(x,y)=0$ if and only if $x=y$ $\delta(x,y)=\delta(y,x)$ for all $x,y\in X$ there exist ...
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Which of the conditions imply that a function is uniformly continuous relative to an uniformity?

Let $(X, \mathcal{U})$ be an uniform space. $f:(X, \mathcal{U})\to (X, \mathcal{U})$ is called uniformly continuous relative to $ \mathcal{U}$, if for every entourage $V\in \mathcal{U}$, $(f\times f)^{...
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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\...
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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 $...
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Induced neighbourhoods in uniform space

I'm largely following Ioan M. James' book for this. Some definitions for a uniform space $X$: For any entourage $U$ of $X$ and any $x \in X$, the neighbourhood induced by $U$ is the set $$ U[x] ...
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Limit point compact uniform space

I'm working on a theorem on compactness for uniform spaces. Here are the definitions I'm using: $X$ is compact if every open cover of $X$ reduces to a finite subcover. $X$ is filter-compact ...
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For every closed neighborhood $\Delta_X\subset D$ , Is there an entourage $U$ with $U\subseteq D$?

Let $(X, \mathcal{U})$ be an uniform space. It is known that every entourage $U\in\mathcal{U}$ is a neighborhood of $\Delta_X$, but the converse is not true, in general. What can say about closed ...
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In discrete topology, every neighborhood of $\Delta_X$ is an entourage?

Let $X=\{x_n\}_{n\in \mathbb{N}}$ where $x_n=\sum_{i=1}^{n}(\frac{1}{i})$ given the metric $d$ inherited from $\mathbb{R}$. Also let $\mathcal{U}(d)$ be uniformity generated by metric $d$. We know ...
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Are uniformly equivalent metrics with the same bounded sets Holder equivalent?

This is a follow-up to my question here.  Let $d_1$ and $d_2$ be two metrics on the same set $X$.  Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $...
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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 $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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Do boundedness in a metric and boundedness in a uniformity not coincide?

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 ...
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For closed set $S$ in uniform space $(X, \mathcal{U})$, with $S\subseteq W$, is there $U\in \mathcal{U}$ such that $S\subseteq U[S]\subseteq W$?

Let $(X, \mathcal{U})$ be a compact, Hausdorff uniform space and $S\subseteq X$ be a closed set with $S\subseteq W$, where $W\subseteq X$ is an open set in $X$. Let $U[x]=\{y: (x, y)\in U\}$ ...
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When is a topological space uniquely uniformizable?

In general, multiple uniformities can induce a given topology. But if a topological space is a compact Hausdorff space, then there is only one uniformity which induces its topology. The formal way ...
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What topological spaces satisfy yet another property involving relatively compact sets?

This is a follow-up to my questions here and here. A subset of a topological space is called relatively compact if its closure is compact. Let's call a sequence $(U_n)$ of open sets a bounding ...
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Are uniformly equivalent metrics with the same bounded sets strongly equivalent?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are ...
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Do the metrics $d$ and $\frac{d}{1+d}$ induce the same uniformity?

Let $d_1$ and $d_2$ be two metrics on the same set $M$. Then $d_1$ and $d_2$ are called uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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Is the notion of Cauchy sequences definable in a bornological topological space?

Being a Cauchy sequence is not a topological property, i.e. two metrics can induce the same topology and yet a sequence which is Cauchy in one may not be Cauchy in the other. It is a uniform property ...
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Is every uniformity for a metrizable topology metrizable?

Let $X$ be a metrizable topological space, and let $U$ be a uniformity which induces the topology on $X$. My question is, is $U$ necessarily metrizable? That is, is $U$ induced by some metric on $X$?...
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When is a bornology on a uniformizable space induced by a uniformity?

Let $X$ be a metrizable topological space, and let $B$ be a nontrivial bornology on $X$. Sze-Tsen Hu showed in 1949 that $B$ is the collection of bounded sets with respect to some metric for the ...
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Clarifying the definition of Boundedness in Uniform Spaces

This excerpt from a journal paper says if X is a uniform space and $A$ is a subset of $X$, then $A$ is said to be bounded if for each entourage $V$, "there exists a finite set $F$ and a positive ...
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Is it true that every perfect set of a compact Hausdorff space is uncountable?

Let $(X, \mathcal{U})$ be a compact Hausdorff space and it has no isolated point. Let $A\subseteq X$ is a closed and infinite set with no isolated point . Is it true that $A$ is uncountable set? ...
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Is there a $D$- chain between two point of a connected component in an uniform space?

A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map of the space into a discrete space is a constant map also a topological space is connected if and only if ...
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Question on $T_1$-topological spaces with compatible uniformities having countable bases

It is known that A $T_1$-topological space is metrizable $\iff$ it has a compatible uniformity with a countable base. Let $(X,\tau')$ be a $T_1$-topological space and $\mathcal U'$ be a ...
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Are the three notions of compactness equivalent for uniform spaces?

A topological space is compact if every open cover has a finite subcover. A topological space is sequentially compact if every sequence has a convergence subsequence. And a topological space is ...
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What is an example of a compact non-uniformizable space?

Every compact Hausdorff space is uniformizable. But I don’t think every compact space is uniformizable. So my question is, what is an example of a compact non-uniformizable space?