# 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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### 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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### 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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### 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 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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### 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 ...
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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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### 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 is an example of a compact non-uniformizable space?

Every compact Hausdorff space is uniformizable. But I don’t think every compact space is uniformizable. So my question is, what is an example of a compact non-uniformizable space?
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### Definition of uniform subspace

I would like some help with the definition of a uniform subspace. The textbook I refer to is Topological and Uniform Spaces by IM James, and he defines the uniform subspace on page 97. Here is his ...
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### Examples of Uniform Spaces

Are there any important examples of uniform spaces other than metric spaces and topological groups? Also, what is an example of when the uniform structure of topological groups is used?
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### quasi-uniformisation of bitopological spaces

By a well-known result of Pervin every topological space is quasi-uniformisable. Since a quasi-uniformity always induces two topologies, one naturally obtains from a quasi-uniform space a ...
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### How to prove the product of totally bounded uniform spaces is totally bounded?

One should note that the family here may not countable. If it is countable, the it is a consequence of the following results; Lemma 1. the product of countable totally bounded metric spaces is ...
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### Let $(X, \mathcal{U})$ be a uniform space and $C\in\mathcal{U}$ be given. Is there a compact set $D\in\mathcal{U}$ such that $D\subseteq C$?

Let $X$ be first countable, locally compact, paracompact, Hausdorff space. We know that $X$ has a uniformity $\mathcal{U}$. Thus $(X, \mathcal{U})$ is a uniform space. Is it true that : For every \$E\...