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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Is every zero-dimensional metrisable space ultrametrisable?

Let us recall a few definitions. A topological space is metrisable if it is homeomorphic to a metric space. ultrametrisable if it is homeomorphic to an ultrametric space. zero-dimensional if every ...
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Understanding mapping from space of uniformly spaced points to non-uniformly spaced points

Given two spaces $X$ and $Y$ with uniformly spaced coordinate system, I can produce a mapping where the new coordinates are non-uniformly spaced. As a simple example, let $X$ = sin($\pi$x) = $Y$, and $...
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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 $...
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Show that any compact uniform space (compact for the topology of the uniformity) is complete

My efforts: Let the uniform space be $(S,\mathcal{U})$. For a Cauchy net {$x_\alpha$}, the collection of all $B_\gamma$ = {$x_\alpha:\alpha\geq\gamma$}, $\gamma\in I$, is a filter base that extends to ...
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A separated uniformity on a set having more than one point never converges as a filter for its product topology

A uniform space $(S,\mathcal{U})$ is called separated iff for every $x\neq y$ in $S$, there is a $U\in\mathcal{U}$ with $\langle x,y\rangle\notin U$. Show that if $S$ has more than one point, a ...
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A uniformity for a space is pseudometrizable iff it has a countable base

Definition. Given a set $S$, a uniformity on $S$ is a filter $\mathcal{U}$ in $S\times S$ with the following properties: (a) Every $A\in \mathcal{U}$ includes the diagonal $D:=\{\langle s,s\rangle:s\...
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A question regarding 'Uniformization' theorem : connection between a topological space and a uniform spaces

This title might be slightly misleading since I saw that the name 'uniformization' theorem is already used in a different context, but my question regards whether a topological space arises from a ...
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Is there a sequence of nested entourages $\{U_n\}$ with $\bigcap U_n=\Delta_X$, if $X$ is compact Hausdorff uniform space

Let $(X, \mathcal{U})$ be a compact Hausdorff uniform space. It is known that $\bigcap \{U: U\in \mathcal{U}\}= \Delta_X$. Is there a sequence $\{U_n\}_{n\in\mathbb{N}}$ with $U_{n+1}\subseteq U_n$ ...
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Motivation for Uniform Cover definition of Uniform Space

The Uniform Cover definition of a Uniform Space is - A uniform space $(X,\Theta)$ is a set $X$ equipped with a distinguished family of uniform covers $\Theta$ from the set of coverings of $X$, ...
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Even covers in Kelley's General Topology

Kelley in his book General Topology introduces a notion of even cover. A cover $\mathcal{U}$ of a topological space $X$ is even if there exists a neighborhood $\mathcal{V} \subset X \times X$ of the ...
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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 \...
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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) $?
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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 ...
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The uniform subspace of X is identical with the uniform subspace

If $ B\subseteq A\subseteq X $, then the uniform subspace $ B $ of $ X $ is identical with the uniform subspace $ B $ of the uniform subspace $ A $ of $ X $ ?
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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^{'} $...
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Topology induced by Uniformity?

Let $f:(X,U) \to (Y,U^{'})$ is a mapping , then $ f $ is uniformly continuous iff $ U^{''}\subset U $ where $ U^{''} $ is the induced uniform structure on $ X $ by $ f $ .
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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 ...
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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 ...
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Uniform Space induced by pseudo metric

Let $X$ and $Y$ be uniform spaces where the uniform structures are induced by pseudo metrics $d$ and $d^{'}$ respectively. Then a function $f:X \to Y$ is uniformly continuous if $\forall \epsilon >...
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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 ...
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Root of an entourage

In a uniform structure we define $$U^{1/n}=\bigcup\{E:E^n\subset U\}$$ We can show that $$(U^{1/n})^m\subset (U^m)^{1/n}\quad\text{; } m>n\Rightarrow U^{1/n}\subset U^{1/m}$$ $$U^n\subset E\...
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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 $X$ ?
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Uniformly continuous in uniform space

Let $(\Bbb R,U)$ be the uniform space induced by the usual metric space $(\Bbb R,d)$ .Show in details that the function $f:(\Bbb R,U)\to (\Bbb R,U)$ , $f(x)=x^3$ is homemorphism, but not uniformly ...
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The space $\omega_1$ of all countable ordinals is not completely uniformizable

I recently heard a lecture where the following fact was mentioned without further comment: The space $\omega_1$ of all countable ordinals is not completely uniformizable. (I.e., there is no ...
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Uniform cover of a uniform space

Let $X$ be a totally bounded uniform space with uniformity $\mathcal{U}$. For an entourage $E$ in $\mathcal{U}$, $\mathcal{F}=\{E[x] : x \in X\}$ is a covering of $X$. Is true that there is a uniform ...
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For entourage $U_0\in\mathcal{U}$, is there a closed entourage $U_1\subseteq U_0$?

Let $(X, \mathcal{U})$ be an uniform space. I know that for closed neighborhood $D$ of $\Delta_X$, there is no an entourage $U\in \mathcal{U}$ with $U\subseteq D$, in general. Is it true for an ...
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Question about Uniform Spaec with a nested space

This question is from the book "General Topology" written by John Kelly and it is Exercise D in Chapter 6, Page 204. For definition of uniform space and the topological generated by the uniform, ...
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Every element of the uniformity include a cube of a symmetric element

The following questions is from the book "General Topology" written by John L.Kelly and located in Page 179 in the proof of Thm 6.6. Given ($X, \tau$), for definition of uniform space built up from $...
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Non-archimedean uniform space

I know about uniform spaces and read this question: A property of uniform spaces Now I am very curious and I would like to learn more about these non-Archimedean uniform spaces. For instance, I am ...
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neighborhood of diagonal $X$

Let $X$ be a topological space and $\Delta_X=\{(x, x): x\in X\}$. Let $D$ be a closed neighborhood of $\Delta_X$ and $D\neq \Delta_X$, is there an open neighborhood $N$ of $\Delta_X$, $N\neq \...
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For entourage $D\in \mathcal{U}$, is there $x_n\in X$ with $\operatorname{Fix}(f)=\bigcup_{n\in\mathbb{N}}D[x_n]\cap\operatorname{Fix}(f)$?

Let $(X, \mathcal{U})$ be a strongly Lindeloff uniform space ( it is complete separable) and $f:X\to X$ be a homeomorphism. A point $x\in X$ is called a fixed point of $f$ if $f(x)=x$. Denote by $Fix(...
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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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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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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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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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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 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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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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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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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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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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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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