The uniform-spaces tag has no wiki summary.
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Generating points in rectangle
I am trying to generate $N$ points randomly and uniformly distributed in an $m \times n$ rectangle. How can this be done? I have tried to split the initial rectangle into as many rectangles i could, ...
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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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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 ...
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Question about the definition of Dieudonné completion
Dieudonné completion $\mu X$ of a space $X$ is the completion of $X$ with
respect to the maximal uniformity $U_X$ on $X$ compatible with the topology of $X$.
I failed to find references to ...
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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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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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Is the fine uniformity of a metric space, metric?
Let $d$ be a metric on the set $X$ and let $\mathcal T$ be it's topology and $\mathcal D$ be the finest uniformity on $X$ which induces $\mathcal T$.
Does $\mathcal D$ have a countable base?!
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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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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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Is every $T_4$ topological space divisible?
A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the ...
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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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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 ...
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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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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,
...
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1answer
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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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Anti-symmetric property of embedding in topological spaces.
$(X,\mathcal T)$ and $(Y,\mathcal S)$ are topological spaces. $X$ can be embedded homeomorphically in $Y$ and $Y$ can be imbedded homeomorphically in $X$.
Are $X$ and $Y$ homeomorphic?
How about ...
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Open sets in a topology produced by metrics.
For each $i\in I$, $$d_i:X^2\to \Bbb R$$ is a metric and $\mathcal T$ is the coarsest topology on $X$ containing all topologies produced by the $d_i$ .
For $U\subseteq X$ we have:
$$\forall x \in ...
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Open sets in uniform and box topology
Let $\mathbb{R}^{\omega}$ denote the product of countably-many copies of $\mathbb{R}$. Let $\bar{d}$ be the following metric on $\mathbb{R}$: $$\bar{d}(x,y)=\inf\{|x-y|,1\}$$
The uniform metric on ...
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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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1answer
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Uniform structure induced by a mapping from its codomain to its domain?
Suppose $X$ is a set and $Y$ is a uniform space with uniform structure $M$.
Given a mapping $f: X \to Y$, I was wondering if $f$ can induce a uniform structure on $X$ from the uniform structure $M$ ...
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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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Hausdorff completion of a uniform space with pseudometrics.
I'm having trouble constructing the Hausdorff completion of a uniform space $(X,U)$ using pseudometrics. I know that every uniformity on a space $X$ is made by pseudometrics. Here is my idea:
Let ...
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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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Every unifom space can be embedded in a product of (pseudo-)metric spaces.
In this book page 18 I found this theorem:
Every uniform space can be embedded in a product of metric spaces.
I googled, I found only a similar theorem about pseudo-metric spaces. I think it ...
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Definition of product uniform space.
Let for each $i\in I$, $(X_i,\mathcal{D}_i)$ be a uniform space. How is the product uniform space defined? Does it produce the product (Tychonoff) topology on $\prod_{i\in I}{X_i}$?
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Is the intersection of two uniformities a uniformity?
Is there any two uniformities $\mathcal{D}_1$ and $\mathcal{D}_2$ on a set $X$ such that $$\mathcal{D}_1\cap \mathcal{D}_2$$ is not a uniformity on $X$?
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Suppose every complete subset of a uniform space is closed. Is it Hausdorff?
Suppose
$(X,\mathcal{D})$ is a uniform space.
for each nonempty subset $A\subseteq X$, if the subspace $(A,\mathcal{D}_A)$ is complete then $A$ is closed.
Is $(X,\mathcal{D})$ Hausdorff?
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1answer
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Counterexample for an Alternate Definition for Uniform Spaces.
Let $\mathcal{D}$ be a filter on $X^2$ such that:
$(\forall D\in\mathcal{D})(\Delta(X)\subseteq D)$.
$(\forall D\in\mathcal{D})(D\circ D\in \mathcal{D})$.
$(\forall D\in\mathcal{D})(D^{-1}\in ...
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a counterexample for Uniform Spaces
Uniform Space is a generalization of metric spaces .
In a uniform space the closure of a singleton $\{x\}$ is the intersection of all neighborhoods of $x$.
Find an infinite topological space such ...
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Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?
And what else can be said, if so?
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
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Is a quotient of a complete group always complete?
Let $\: \langle G,\cdot,\mathcal{T}\hspace{0.01 in} \rangle \:$ be a $\big($$\text{T}_0$$\big)$ topological group. $\;\;$ Let $H$ be a closed normal subgroup of $G$.
Set $\;\; \mathbf{G} \: = \: ...
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Pointfree generalization of uniform spaces?
Topological spaces generalize as frames and locales.
But are there a pointfree generalization of uniform spaces?
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Uniform covers and partitions of unity
A cover $\mathcal C$ of a uniform space $(X,\mathcal U)$ is called a uniform cover if there is $U\in \mathcal U$ such that the cover $\{U(x):x\in X\}$ refines $\mathcal C$.
Is it true that to 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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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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Definition of product of uniform spaces
In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous.
But Springer's encyclopedia ...
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Extension of pseudometrics to Hausdorff completion
Let $(X,\mathcal{U})$ be a uniform space with Hausdorff completion $(X',\mathcal{U}')$ (made by the minimal Cauchy filters). Since $X$ is uniform, $\mathcal{U}$ is generated by pseudometrics ...
