# 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).

242 questions
Filter by
Sorted by
Tagged with
23 views

### 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 ...
22 views

17 views

### 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 ...
14 views

### 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 ...
18 views

21 views

### 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 $d(f\times f)$?
35 views

### 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 ...
21 views

### 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 $X$ ?
17 views

### 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 $X^{'}$...
32 views

### 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 $f$ .
21 views

### 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 ...
35 views

### 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 ...
19 views

50 views

### 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 ...
54 views

### 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 ...
90 views

### 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-...
54 views

### 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 ...
135 views

### 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 ...
94 views

### 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 ...
38 views

### 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 ...
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 $D\circ D \... 1answer 33 views ### Let$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$?... 0answers 68 views ### Hausdorff completion of uniform spaces I'm working on constructing a completion of uniform spaces. As far as possible I'd like to follow the construction in the case of metric spaces that uses equivalence classes of Cauchy sequences ... 1answer 137 views ### Are metrics uniformly equivalent if and only if they have the same zero-distance sets? Let$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)$are ... 1answer 73 views ### Relation Between Nbhd Base at$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$, ... 0answers 19 views ### Is there a proper set in a locally compact, paracompact uniform space? Let$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$is the family of all ... 1answer 25 views ### Is there an entorage$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$, but the converse of it, is not true. For instance, Consider$...
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)^{...