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).

Filter by
Sorted by
Tagged with
0 votes
2 answers
36 views

Does every net has a convergent subnet in a compact uniform space?

We know that Result 1: In a metric space compactness implies sequential compactness, that is, if $(X,d)$ is a compact metric space then every sequence $\{x_n\}$ in $X$ has a convergent subsequence. In ...
user avatar
  • 25
0 votes
0 answers
27 views

Delone sets in terminology of uniformities

I've been reading about Delone sets recently, and noticed that they are formulated either in terms of topological groups or in terms of metric spaces. Group terminology: Let $G$ be a locally compact ...
user avatar
  • 5,329
1 vote
0 answers
24 views

Contraction principle type theorems for uniform spaces

I was recently wondering whether there are conraction principle like theorems for uniform spaces but which are not metric spaces. The theorem relies heavily on the notion of completeness, which exists ...
user avatar
  • 5,329
0 votes
1 answer
25 views

Certain properties of uniform structure

I am trying to read about unifrom spaces from Introduction to Uniform Spaces, and I was wondering about some basic facts which I wasn't able to find there. Let $(X,\mathcal{E})$ be a uniform space. ...
user avatar
  • 5,329
2 votes
1 answer
25 views

Entourage asymmetry and Cauchy nets: $x_a$ is $V$-close to $x_b$ does not imply $x_b$ is $V$-close to $x_a$

I have a few quibbles about the nature of the ordering $(a,b)$ versus $(b,a)$ when it comes to membership of an entourage in a uniform space. The Wikipedia article on uniform spaces nowhere asserts ...
user avatar
  • 8,404
3 votes
0 answers
40 views

Quibble with proofs of the compact-open topology being equivalent to the uniform convergence topology: what happens if $f(x)$ is an isolated point?

Let $X$ be a topological space and $Y$ a uniform space. I would like to understand the proof that, if one endows $Y^X$ with the compact-open topology, if $f_n\to f\in Y^X$ under this topology then $...
user avatar
  • 8,404
2 votes
0 answers
58 views

What are uniform spaces actually useful for?

I've used to study about uniform spaces from the book of J. R. Isbell named simply "Uniform Spaces". Isbell defines them using something called uniform covers (confusingly calling them ...
user avatar
  • 5,732
0 votes
0 answers
13 views

Proximity to an arbitrary family and compact space

https://en.wikipedia.org/wiki/Proximity_space If we know that $A$ is not proximity to $A_\lambda$, $\forall\lambda\in\Lambda$. Is it necessarily true that $A$ is not proximity to $\cup_{\lambda\in\...
user avatar
  • 1,811
0 votes
1 answer
28 views

Neighborhood base generated by uniformity

In Topology for analysis by A. Wilansky the theorem 11.1.2 states: Let $(X,\mathcal{U})$ be a uniform space, let $\mathcal{B}$ be [uniformity] base for $\mathcal{U}$ and let $x \in X$. Then $\{U(x) :...
user avatar
  • 1,779
1 vote
0 answers
69 views

Intuition for a uniform space?

Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. According to Wikipedia, ...
user avatar
1 vote
0 answers
59 views

Necessary and suficient condition for a uniform space to be complete

I am trying to learn a bit about linear topological rings and modules. In doing that, I have stumbled upon some questions about uniform spaces and I am struggling to find references. My main is ...
user avatar
  • 366
1 vote
2 answers
37 views

How to prove that there is an entourage $V$ such that $V(B) \cap A = \emptyset$ in a uniform space when A is compact and B is closed?

I am studying Michael's article "Topologies on Spaces of Subsets" and I just can't figure out the proof of lemma 2.2.2' below I have already proven that if $[X,U]$ is $T_1$ (which we are ...
user avatar
  • 117
0 votes
1 answer
40 views

Where can I find more insight about spaces of subsets of a base space?

I've been studying Michael's article "Topologies on spaces of subsets" and he states some propositions and lemmas without proving, asserting that they follow directly from the definitions ...
user avatar
  • 117
0 votes
0 answers
31 views

How to prove in metric space that classical definition of the Cauchy sequence coincides with the entourage definition of the Cauchy sequence?

Let $\{a_n\}$ denote sequence in metric space X. We have well known classical definition of Cauchy sequence: $ \forall_{\epsilon>0}\exists_N\forall_{n,m>N}d(a_n,a_m)<\epsilon $ We have also ...
user avatar
0 votes
1 answer
95 views

Why are uniform spaces defined the way they are? What is uniformity saying intuitively?

I have recently learned about uniform spaces. My intuition is that uniformity is about notion of closeness (points can be quite close, or very close, or very very close.. to each other). However, I ...
user avatar
4 votes
1 answer
66 views

Motivation and references for 'the topology of uniform convergence'

I'm a graduate student working on $\textbf{algebraic number theory}$. While reading papers, I have seen authors mentioning that the group of characters is equipped with the topology of uniform ...
user avatar
  • 183
0 votes
1 answer
26 views

Different definitions of quotient maps for uniform spaces 2

Here I asked a question about $3$ definitions of quotients for uniform spaces being equivalent (uniform space here, is a Hausdorff space together with uniform covers - those can be found on wikipedia)....
user avatar
  • 5,732
0 votes
1 answer
37 views

Different definitions of quotient maps for uniform spaces

I have the following $3$ definitions of a quotient map of uniform spaces: A surjective morphism $q:X\to Q$ such that if $q = g\circ f$ where $g, f$ are morphisms and $g$ is a bijection then $g$ is an ...
user avatar
  • 5,732
0 votes
0 answers
7 views

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 ...
user avatar
  • 5,732
2 votes
0 answers
55 views

Non-trivial metric space $X$ with all of $C(X)$ uniformly continuous.

Is there a metric space $X$ such that any $f\in C(X)$ is uniformly continuous, but there is a metric space $Y$ and $g:X\to Y$ which is continuous but not uniformly continuous? If $X$ is compact or ...
user avatar
  • 5,732
0 votes
0 answers
33 views

Existence of smallest preuniformity containing a given cover.

Let $X$ be a set and $\mathcal{U}$ a cover of this set. A preuniformity $\mu$ on $X$ is a family of covers with the following conditions satisfied: $\{X\}\in \mu$ Any cover in $\mu$ has a star-...
user avatar
  • 5,732
0 votes
0 answers
20 views

The eigenfunctions of $T_{\phi}f:=f\circ\phi$ span $C(X)$ if and only if $\{\phi^{n}:n\in\mathbb{Z}\}\subset C(X,X)$ is equicontinuous.

Let $X$ be a (not necessarily metrizable) compact Hausdorff space. We endow $C(X,X)$ with the topology of uniform convergence. Let $\phi\colon X\to X$ be a homeomorphism and consider the isometric ...
user avatar
  • 2,958
4 votes
0 answers
53 views

If the eigenfunctions of $T_{\phi}f:=f\circ\phi$ span $C(X)$, then $\overline{\{\phi^{n}:n\in\mathbb{Z}\}}^{\text{unif}}$ is compact

Let $X$ be a compact Hausdorff space. We endow $C(X,X)$ with the topology of uniform convergence. Let $\phi\colon X\to X$ be a homeomorphism and consider the linear isomorphism $T_{\phi}\colon C(X)\to ...
user avatar
  • 2,958
-1 votes
1 answer
65 views

Hausdorff space which is not Urysohn space.

Let $X=\mathbb{R}\cup \{\infty\}$ and for every $p\in \mathbb{R}-\{0\}$, $p$ has the usual neighborhoods in $\mathbb{R}-\{0\}$. Basic neighborhoods of $0$ is the sets of the form $\{[0, \frac{1}{n}):n\...
user avatar
1 vote
1 answer
42 views

Topology of uniform convergence of functions $X\to X$?

Suppose that $X$ is a compact Hausdorff space and that $\phi\colon X\to X$ is a homeomorphism. What does it mean to take the closure of $\{\phi^{k}:k\in\mathbb{Z}\}$ with respect to the topology of ...
user avatar
  • 2,958
0 votes
1 answer
41 views

Uniform Structure on the Space of Subsets (Hyperspaces)

I'm studying the article "Topology on Spaces of Subsets" by Ernest Michael but I can't understand how he defines the uniform structure on on the space of non-empty closed subsets (a ...
user avatar
  • 117
0 votes
0 answers
35 views

What is a good book for uniform spaces using pseudometrics definition?

I want to learn uniform spaces. I like pseudometrics definition. Is there a book on uniform spaces using pseudometrics definition? The famous the better. I know most of the core subjects in ...
user avatar
  • 1,235
1 vote
0 answers
17 views

For entourage $U$ and continuous map $t:X\to X$, is there an entourage $V$ with $t^{-1}V[x]\subseteq U[x]$?

Let $(X, \mathcal{U})$ be a uniform space, $T$ be a topological semigroup and $(T, X)$ be a semiflow. This means that $t:X\to X$ is continuous and $(t_0t_1)(x)=t_0(t_1x)$ and if $e\in T$, then $ex=x$. ...
user avatar
5 votes
1 answer
107 views

A property of compact metric spaces

I would like to know if the following property could hold. Let $(X,d)$ be a compact metric space. Then for every $\epsilon >0$ there exists some $\delta >0$ and some continuous function $F:X\...
user avatar
  • 126
0 votes
1 answer
19 views

Does normal space guarantees existence of normal cover

Let $X$ be a topological space and $\mathcal{U}$ be an open cover of $X$. Define $st(M)=\{ U\in\mathcal{U}\vert M\cap U \neq \emptyset \}$ where $M$ is an arbitrary subset of $X$. A cover $\mathcal{V}$...
user avatar
  • 3,106
0 votes
1 answer
55 views

Alternative characterisation of weakly complete Banach spaces

Let $V$ be a topological vector space over some locally compact field $\Bbb K$. Let $V'$ denote all continuous functionals on $V$, the weak topology is defined to be the coarsest topology on $V$ ...
user avatar
  • 19.7k
0 votes
0 answers
82 views

Topology of local uniform convergence

Let X be a topological space and Y a uniform space. Does there exist a topology of local uniform convergence on $Y^X$?
user avatar
  • 105
0 votes
1 answer
135 views

Equivalence of metric

Definition 1. Two metrics $d_{1}$, $d_{2}$ on a set $X$ are equivalent, if there exist two positive numbers $c$, $c'\in\mathbb{R}_{> 0}$, such that if $x$, $y\in X$ then$$ c d_{2}(x, y)\leq d_{1}(...
user avatar
  • 129
0 votes
1 answer
23 views

Relation between entourages and finite open covers of compact Hausdorff spaces

Let $X$ be a compact Hausdorff space and $\mathcal{V}$ be a finite open cover of $X$. Since $X$ is a compact Hausdorff space, hence it has a uniformity $\mathcal{U}$. Is there an entourage $D$ in ...
user avatar
0 votes
1 answer
49 views

Definitions of uniform space

Let $X$ be a topological space. The collection $\mathcal{A}=\{U_\lambda(x)\}_{\lambda\in \Lambda, x\in X}$, with $U_\lambda(x)\subseteq X$, is called a uniform structure of $X$ if If $x\in X$ and $\...
user avatar
6 votes
1 answer
163 views

What happened to bornological/uniform spaces?

When reading older papers, I often see references to bornological or uniform spaces, which encode the notions of "boundedness" or "uniformness". In this way, they seem to sit ...
user avatar
0 votes
1 answer
23 views

A question about normal space and uniform space

Let $X$ be a compact and Hausdorff space. Hence there is a uniformity $\mathcal{U}$ on $X$, also $X$ is normal space i.e. for every point $x\in X$ and every closed set $A\subseteq X$ there exist open ...
user avatar
0 votes
1 answer
77 views

Intersection of nested open sets in compact Hausdorff space

Let $U$ be an open subset of compact Hausdorff space $X$. Choose a non-empty subset $U_1$ in $X$ with $\overline{U}_1\subseteq U$. Repeating this construction without end, we can then find a sequence ...
user avatar
1 vote
1 answer
62 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 ...
user avatar
  • 36.1k
0 votes
0 answers
33 views

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 $...
user avatar
  • 103
0 votes
1 answer
42 views

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 $...
user avatar
  • 5,732
0 votes
1 answer
35 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 ...
user avatar
0 votes
1 answer
23 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 ...
user avatar
0 votes
1 answer
37 views

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\...
user avatar
0 votes
1 answer
29 views

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 ...
user avatar
  • 5,329
1 vote
1 answer
32 views

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 $U_{n+1}\subseteq U_n$ ...
user avatar
0 votes
1 answer
43 views

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 $X$, ...
user avatar
  • 3,341
1 vote
1 answer
88 views

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 ...
user avatar
  • 1,905
0 votes
1 answer
40 views

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 \...
user avatar
0 votes
1 answer
35 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) $?
user avatar

1
2 3 4 5 6