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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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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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a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
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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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Is $\operatorname{Homeo}([0,1])$ Weil-Complete?

After learning about uniformities on topological groups, we were given several sources to read. I came across the term "Weil-complete." A topological group is Weil-complete if it is complete with ...
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Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
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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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When is a uniform space complete

From Wikipedia: a uniform space is called complete if every Cauchy filter converges. I was wondering if the following three are equivalent in a uniform space: every Cauchy filter converges, every ...
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Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It ...
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non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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Do $\Bbb H$ and $\Bbb R^2$ have the same uniform structure?

The hyperbolic upper half plane $\Bbb H$ and euclidean space $\Bbb R^2$ are not isomorphic as metric spaces, which can be see from the fact that in $\Bbb R^2$ for any point not on a geodesic there ...
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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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non-hausdorff completion of a uniform space.

Let $(X,\mathcal U)$ be a Hausdorff uniform space. Can $(X,\mathcal U)$ have a non-hausdorff completion?
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Separation axioms in uniform spaces

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ ...
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Compactly supported continuous function is uniformly continuous.

What is the most general space where compactly supported continuous functions are uniformly continuous? I managed to prove this for metric spaces but I am interested if it also holds in more general ...
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Spaces with the property: Uniformly continuous equals continuous

I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces. This fact inspires me to ask: Are there ...
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Is a quotient of a complete group always complete?

Let $\: \langle G,\cdot,\mathcal{T}\hspace{0.01 in} \rangle \:$ be a $\big($$\text{T}_0$$\big)$ topological group. $\;\;$ Let $H$ be a closed normal subgroup of $G$. Set $\;\; \mathbf{G} \: = \: \...
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Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta X^2\subseteq\...
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$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
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Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?

AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
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Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
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Analog of an open map for uniform structures

In topology, a function is continuous if the inverse image of an open set is open. A function is open if the image of an open set is open. Uniformity continuity can be defined in a similar way as ...
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Showing a uniformity is complete.

I've seen in various textbooks and notes that if $X$ is paracompact, then the collection of all the neighborhoods of the diagonal is a uniformity. I am trying to show that this uniformity is complete ...
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Open sets in uniform and box topology

Let $\mathbb{R}^{\omega}$ denote the product of countably-many copies of $\mathbb{R}$. Let $\bar{d}$ be the following metric on $\mathbb{R}$: $$\bar{d}(x,y)=\inf\{|x-y|,1\}$$ The uniform metric on $\...
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Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?

And what else can be said, if so? In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
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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 $\overline{D}\subseteq D\circ D$ in a uniform space?

Suppose $(X,\mathcal{D})$ is a uniform space and $D\in\mathcal{D}$. Is it true that $$\overline{D}\subseteq D\circ D,$$? here we use the product topology to define $\overline{D}$.
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Benefits from using the uniform structure of compact Hausdorff spaces

It is well known that every compact Hausdorff space admits a unique (necessarily complete) uniform structure which is compatible with the topology, and every continuous function from such a space to a ...
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When does pointwise convergence on compact space imply uniform convergence?

I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then ...
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Definitions and coincidences of the topology of pointwise convergence and the uniformity of uniform convergence

I was wondering how "the topology of pointwise convergence" is defined on $Y^X$ where $X$ is a set and $Y$ is a topological space? Are there more than one topologies that can topologize pointwise ...
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“Uniform groups” (similar to topological groups)?

Why have I heard about topological groups, but nothing about "uniform groups" (uniform spaces endowed with a group)?
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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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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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Is the long line completely uniformizable?

The long line $L$ is uniformizable; in fact, as $L$ is a locally compact Hausdorff space we can explicitly write down a uniformity for it: If $\hat{L}$ is the one-point compactification of $L$, then $\...
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Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
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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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Completion of a linear order that is a dense subspace of a compact space.

Suppose $D$ is a linearly ordered space which is densely embedded in a compact Hausdorff space $K$. What can we say about the relation between $K$ and $\overline D$, the completion of $D$. Is one a ...
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Is $\approx$ actually an entourage?

I was looking at applying the ideas in the paper On Nonstandard Topology to Uniform spaces. Given a uniform space $(X,\Phi)$, we can define the relation $\approx$ on ${}^*X$ as follows $$\approx \, = \...
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Existence of an unique uniform space, generated by a family of coverings.

It is known, that given a family $M$ of subsets of a set $X$, there exists a unique, minimal topology $\tau$, such that $M \subset \tau$, we get this topology as the intersection of all topologies ...
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Boundedness in uniform spaces?

After looking a bit at uniform spaces, as their general definition seems relevant to the study of topological vector spaces, it seems that they provide just enough structure to define the notion of ...
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If $\mathcal{B}$ in $X$ converges to $x$, it accumulates at $x$ and if $X$ is Hausdorff, $x$ is a unique point of accumulation.

Essentially I need feedback, mostly on writing style and accuracy and tightness of arguments. Suppose $\mathcal{B}$ converges to $x$. For every open neighbourhood $U(x)$ of $x$ in $X$ there exists $...
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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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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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Why are locally compact groups Weil complete?

Why are locally compact groups Weil complete? Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent. Thank you, and sorry if I have bad writing.
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Completeness of Continuous Functions on Uniform Spaces

So I'm trying to find the most general setting in which I can talk about completeness of function spaces. In metric spaces, it's simple to show that the space $C(X,Y)$ of continuous functions $X\...
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On the Compact Uniformization Theorem

I just read through a proof of the Compact Uniformization Theorem, and I follow it up to the very last line. The proof is: Compact Uniformization Threorem. If $X$ is a compact regular space, then the ...
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Dieudonné complete and topologically complete are equivalent for every space $X$.

How can we show that: For every topological space $X$ the following conditions are equivalent: A space $X$ is topologically complete if $X$ is homeomorphic to a closed subspace of a product of ...
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Is a compact Hausdorff uniform space fine?

Let $\mathcal D_1$ and $\mathcal D_2$ be two uniformities on $X$ which produce the same topologies on $X$ (say $\mathcal T= \mathcal T _{\mathcal D_1}=\mathcal T _{\mathcal D_2}$). If $(X,\mathcal T)$...
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What is the uniform structure on a topological vector space?

I've read that topological vector spaces are canonically uniform spaces, however the definitions of uniform space are quite cryptic to me! Is there a simple/intuitive way to understand why we have ...
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How are Uniform Spaces Hausdorff?

Definition: Let $X$ be a set. A set $U\subseteq X\times X$ is called entourage of the diagonal if $\Delta=\{(x,x):x\in X\}\subseteq U$ and $U=U^{-1}.$ Let $\Phi$ be a family of entourages of the ...
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Neighborhood of diagonal

Consider a uniform space $X$ (with induced topology). What of the following can be a subset of the other? Which of the following is always a subset of the other? Neighborhood of the diagonal in the ...