# 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).

214 questions
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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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### 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 $G$, ...
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.