# 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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### How can we define the supremum norm on the space of continuous functions if they're not necessarily bounded?

I frequently see the uniform norm described as a norm on the space of continuous functions $C(Y, X)$ where $Y$ is a topological space and $X$ is a metric space, especially in the context of the ...
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### Entourage definition to pseudometrics definition of uniform spaces

If $X$ is a set equipped with a collection $(d_i)_{i\in I}$ of pseudometrics, then the corresponding uniform structure (collection of entourages) $\Phi$ is defined by declaring that $U\in\Phi$ if and ...
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### 'Projection' to a closer point on boundary in uniform spaces

I'm considering the following problem over uniform space which seems like it should hold, but I can't seem to show it. Let $X$ be a connected uniform space with a uniformity $\mathcal{E}$. I denote ...
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### Can all the entourages of a uniform structure be transitive?

A uniform structure $\Phi$ on a set $X$ is a collection of relations that satisfy some special properties. In general, all relations must be reflexive. For metric spaces all the relations are ...
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### Is a filter generated by a Dedekind cut a minimal Cauchy filter and vice versa?

I'm reading The Reals as Rational Cauchy filter by I. Weiss and I am trying to establish a connection between Dedekind reals and Bourbaki reals. I'm working with this definition of a Dedekind real a.k....
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### Does every net has a convergent subnet in a compact uniform space?

We know that Result 1: In a metric space compactness implies sequential compactness, that is, if $(X,d)$ is a compact metric space then every sequence $\{x_n\}$ in $X$ has a convergent subsequence. In ...
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### Contraction principle type theorems for uniform spaces

I was recently wondering whether there are conraction principle like theorems for uniform spaces but which are not metric spaces. The theorem relies heavily on the notion of completeness, which exists ...
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### Certain properties of uniform structure

I am trying to read about unifrom spaces from Introduction to Uniform Spaces, and I was wondering about some basic facts which I wasn't able to find there. Let $(X,\mathcal{E})$ be a uniform space. ...
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### Entourage asymmetry and Cauchy nets: $x_a$ is $V$-close to $x_b$ does not imply $x_b$ is $V$-close to $x_a$

I have a few quibbles about the nature of the ordering $(a,b)$ versus $(b,a)$ when it comes to membership of an entourage in a uniform space. The Wikipedia article on uniform spaces nowhere asserts ...
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### Does normal space guarantees existence of normal cover

Let $X$ be a topological space and $\mathcal{U}$ be an open cover of $X$. Define $st(M)=\{ U\in\mathcal{U}\vert M\cap U \neq \emptyset \}$ where $M$ is an arbitrary subset of $X$. A cover $\mathcal{V}$...
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### Alternative characterisation of weakly complete Banach spaces

Let $V$ be a topological vector space over some locally compact field $\Bbb K$. Let $V'$ denote all continuous functionals on $V$, the weak topology is defined to be the coarsest topology on $V$ ...
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### Topology of local uniform convergence

Let X be a topological space and Y a uniform space. Does there exist a topology of local uniform convergence on $Y^X$?
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### Equivalence of metric

Definition 1. Two metrics $d_{1}$, $d_{2}$ on a set $X$ are equivalent, if there exist two positive numbers $c$, $c'\in\mathbb{R}_{> 0}$, such that if $x$, $y\in X$ then c d_{2}(x, y)\leq d_{1}(...
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### Relation between entourages and finite open covers of compact Hausdorff spaces

Let $X$ be a compact Hausdorff space and $\mathcal{V}$ be a finite open cover of $X$. Since $X$ is a compact Hausdorff space, hence it has a uniformity $\mathcal{U}$. Is there an entourage $D$ in ...
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### A question regarding 'Uniformization' theorem : connection between a topological space and a uniform spaces

This title might be slightly misleading since I saw that the name 'uniformization' theorem is already used in a different context, but my question regards whether a topological space arises from a ...
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### Is there a sequence of nested entourages $\{U_n\}$ with $\bigcap U_n=\Delta_X$, if $X$ is compact Hausdorff uniform space
Let $(X, \mathcal{U})$ be a compact Hausdorff uniform space. It is known that $\bigcap \{U: U\in \mathcal{U}\}= \Delta_X$. Is there a sequence $\{U_n\}_{n\in\mathbb{N}}$ with $U_{n+1}\subseteq U_n$ ...
The Uniform Cover definition of a Uniform Space is - A uniform space $(X,\Theta)$ is a set $X$ equipped with a distinguished family of uniform covers $\Theta$ from the set of coverings of $X$, ...