# Tagged Questions

In the mathematical field of topology, a uniform space is a set with a uniform structure. (Def: http://en.m.wikipedia.org/wiki/Uniform_space)

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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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### 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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### Existence of an unique uniform space, generated by a family of coverings.

It is known, that given a family $M$ of subsets of a set $X$, there exists a unique, minimal topology $\tau$, such that $M \subset \tau$, we get this topology as the intersection of all topologies ...
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It's my first post on this site, so please forgive me for all mistakes I've made during writing it. I can't understand the proof of this lemma: Let $( X,\mathcal U )$ be an uniform space, $x\in X$ ...
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### Weaker Than The Weak Topology?

The weak topology on a Banach Space $E$ is defined to have sub-base consisting of open balls of the form $B_\alpha(x,r) = \lbrace y \in E : \vert \alpha(x-y) \vert < r\rbrace$ for each $x \in E$ ...
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### Equivalent definitions of complete uniform space

A uniform space $(X,\mathcal{U})$ is called complete if every Cauchy filter converges. In this page, Brian M.Scott says that, In a uniform space every Cauchy filter converges iff every Cauchy ...
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### When does pointwise convergence on compact space imply uniform convergence?

I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then ...
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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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### Different uniform spaces having the same set of Cauchy filters

I want to understand how Cauchy space is different than uniform space. For this I need an example: An example of two different uniform spaces having for both of them the same set of Cauchy filters?
Consider a uniform space $X$ (with induced topology). What of the following can be a subset of the other? Which of the following is always a subset of the other? Neighborhood of the diagonal in the ...