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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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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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When is a bornology on a uniformizable space induced by a uniformity?

Let $X$ be a metrizable topological space, and let $B$ be a nontrivial bornology on $X$. Sze-Tsen Hu showed in 1949 that $B$ is the collection of bounded sets with respect to some metric for the ...
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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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What topological spaces satisfy yet another property involving relatively compact sets?

This is a follow-up to my questions here and here. A subset of a topological space is called relatively compact if its closure is compact. Let's call a sequence $(U_n)$ of open sets a bounding ...
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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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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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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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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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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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A uniformly continuous function between totally bounded uniform spaces

Let $X$ and $Y$ is a uniform spaces. Let $f$ is a uniformly continuous surjective function $X\rightarrow Y$. Conjecture: If $X$ is totally bounded then $Y$ is also totally bounded.
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TVS: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise to ...
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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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What are interesting properties of totally bounded uniform spaces?

I work on reloids, a generalization of uniform spaces. As such I am interested about properties of totally bounded uniform spaces. My question: What are interesting properties of totally bounded ...
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Do the metrics $d$ and $\frac{d}{1+d}$ induce the same uniformity?

Let $d_1$ and $d_2$ be two metrics on the same set $M$. Then $d_1$ and $d_2$ are called uniformly 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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How can we construct *Fine uniformities*?

Given a uniformizable (w.r.t. entourage uniformity) space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. A uniform ...
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Are uniformly equivalent metrics with the same bounded sets strongly equivalent?

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 ...
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Covering uniformity (Topology)

If X is any uniformizable topological space, then we can show that there exists a finest covering uniformity on X by taking the union of all covering uniformities compatible with the topology of X. ...
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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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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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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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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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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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Are the three notions of compactness equivalent for uniform spaces?

A topological space is compact if every open cover has a finite subcover. A topological space is sequentially compact if every sequence has a convergence subsequence. And a topological space is ...
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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 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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Question about totally disconnected compact uniform space

Let $(X, \mathcal{U})$ be a compact uniform space. For $D\in\mathcal{U}$, the sequence $\{x_i\}_{i\in\mathbb{Z}}$ is called a $D$- chain if $(x_i, x_{i+1})\in D$ for all $i\in\mathbb{Z}$. Let $X$ be ...
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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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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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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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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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Question about neighborhood of diagonal of a topological space

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 \...
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Bounded uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
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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 ...
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Do boundedness in a metric and boundedness in a uniformity not coincide?

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 $F$. A subset of a metric space is said to ...
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The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite ...
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closure of bounded set in uniform space

I have stumbled upon the following statement but I could not see why it is true. Let $(X,V)$ be a uniform space and $A$ bounded in it. Bounded means that for each $W\in V $ there is a finite set $F\...
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precompactness and boundedness in uniform space

Consider a uniform space $(X,\mathscr{U})$. For an entourage $U\in\mathscr{U}$, one says that a set $M$ is small of order $U$, if $M\times M\subseteq U$. $P\subset X$ is precompact if for every $U\in\...
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$\iff$ Condition for metrizable uniformity

Let $X$ be a nonempty set and $\mathcal U$ be a uniformity on $X.$ I know that $\mathcal U$ is said to be metrizable if $\exists$ a metrix $d$ on $X$ such that the uniformity $\mathcal U(d)$ induced ...
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Uniformity inducing Topology

If $\mu$ is any covering uniformity on a set $X$ inducing the topology $\mathcal{T}$ on $X$ and if $(\mathcal{B_n})_{n \in \mathbb{N}}$ is a normal sequence of open covers of $X$ then $\mu \cup \{\...
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a compact uniform space which is not metrizable

Let $(X, \mathcal{U})$ be a compact uniform space which is not metrizable and and $\{U_i\}_{i=0}^{\infty}$ be a countable set of $U_i\in \mathcal{U}$ with $U_{i-1}\subseteq U_i$ and $U_1\subseteq V$ ...