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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How can we define the supremum norm on the space of continuous functions if they're not necessarily bounded?
I frequently see the uniform norm described as a norm on the space of continuous functions $C(Y, X)$ where $Y$ is a topological space and $X$ is a metric space, especially in the context of the ...
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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 ...
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'Projection' to a closer point on boundary in uniform spaces
I'm considering the following problem over uniform space which seems like it should hold, but I can't seem to show it.
Let $X$ be a connected uniform space with a uniformity $\mathcal{E}$. I denote ...
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Can all the entourages of a uniform structure be transitive?
A uniform structure $\Phi$ on a set $X$ is a collection of relations that satisfy some special properties.
In general, all relations must be reflexive.
For metric spaces all the relations are ...
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Is a filter generated by a Dedekind cut a minimal Cauchy filter and vice versa?
I'm reading The Reals as Rational Cauchy filter by I. Weiss and I am trying to establish a connection between Dedekind reals and Bourbaki reals.
I'm working with this definition of a Dedekind real a.k....
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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 ...
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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 ...
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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.
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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 ...
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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 $...
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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 ...
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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\...
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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) :...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)....
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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 $ex=x$. ...
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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\...
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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 $\mathcal{V}$...
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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 $V$ ...
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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 $Y^X$?
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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}(...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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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