# 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).

211 questions
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### Covering characterization of Metrizability and Paracompactness

I am quite struck by the similarity between covering charachterization of Paracompactness and Metrizability both in the T1 and T3 topologies. In particular in the case of T1 topologies we have the ...
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### $\Delta_X$ is $\mu$- isolated set

Let $g:X\times X\to X\times X$ be a continuous function and $(X, \mathcal{U})$ be a compact uniform space. Also for entourage $D\in\mathcal{U}$ suppose \{z\in X\times X: g^n(z)\in D,...
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### Proof verification: totally bounded iff every filter has a cauchy superfilter

It would be appreciated if someone could check if these proofs are correct (in Schechter's HAF, the 'only if' direction of the theorem below is essentially left as an exercise with a hint, so I wanted ...
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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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### Quasi-uniform spaces are not regular in general; where does this argument fail?

It is well known that every topological space is quasi-uniformizable, and every quasi-uniform space $(X, \mathcal U)$ induces a natural topology on $X$. Moreover, if $(X, \mathcal U)$ is a quasi-...
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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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### 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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### For closed set $S$ in uniform space $(X, \mathcal{U})$, with $S\subseteq W$, is there $U\in \mathcal{U}$ such that $S\subseteq U[S]\subseteq W$?

Let $(X, \mathcal{U})$ be a compact, Hausdorff uniform space and $S\subseteq X$ be a closed set with $S\subseteq W$, where $W\subseteq X$ is an open set in $X$. Let $U[x]=\{y: (x, y)\in U\}$ ...
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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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### 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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### 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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### 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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### 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 every uniformity for a metrizable topology metrizable?

Let $X$ be a metrizable topological space, and let $U$ be a uniformity which induces the topology on $X$. My question is, is $U$ necessarily metrizable? That is, is $U$ induced by some metric on $X$?...
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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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### Clarifying the definition of Boundedness in Uniform Spaces

This excerpt from a journal paper says if X is a uniform space and $A$ is a subset of $X$, then $A$ is said to be bounded if for each entourage $V$, "there exists a finite set $F$ and a positive ...
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### Is it true that every perfect set of a compact Hausdorff space is uncountable?

Let $(X, \mathcal{U})$ be a compact Hausdorff space and it has no isolated point. Let $A\subseteq X$ is a closed and infinite set with no isolated point . Is it true that $A$ is uncountable set? ...
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### Is there a $D$- chain between two point of a connected component in an uniform space?

A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map of the space into a discrete space is a constant map also a topological space is connected if and only if ...
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### Question on $T_1$-topological spaces with compatible uniformities having countable bases

It is known that A $T_1$-topological space is metrizable $\iff$ it has a compatible uniformity with a countable base. Let $(X,\tau')$ be a $T_1$-topological space and $\mathcal U'$ be a ...