# 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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### Redundancy in the definition of uniform spaces

Let me paraphrase Wikipedia's definition of uniform spaces: Definition A. A set $X$ endowed with a nonempty collection $\Phi$ of subsets of $X \times X$ is a uniform space if the following conditions ...
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### In a connected compact Hausdorff space two disjoint non-empty closed sets are "joined" by a connected set

This is Exercise 16 of §4 of Chapter 2 of Bourbaki's General Topology. The proof sketched in the hints is quite involved and I can't manage to fill in the details. It's also hard to explain where I'm ...
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### Conditions so that every compact subset of the image is the image of a compact subset of the domain

Question. Let $X$ be a uniform space, and $Y$ Hausdorff uniform, both complete. Then, for a continuous surjection $f: X\to Y$, under what conditions can we claim: for every compact set $K\subset Y$, ...
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### Total boundness & precompactness on Metric Space

Hi i'm studying an introduction course to Functional Analysis and i'm little confusing about the relation between the Total Boundness and Precompactness of space on world of Metric Spaces For clarity ...
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### Proof that the initial uniformity induced by totally bounded spaces is totally bounded

Bourbaki proves this claim in General Topology, chapter II, §4.2, Proposition 3, using the Hausdorff completion. It is however possible to give an "elementary" proof in the case of products (...
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### What is the universal/fine uniformity on a topological group?

I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\langle x,y\rangle:x^{-1}y\in U\text{ and }xy^{-1}\in U\}$ is an ...
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### I.M.James' proof of "uniformizable spaces are completely regular"

In I.M. James' Topological and uniform spaces, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular. The outline ...
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### What properties of metric spaces are not preserved by uniformly continuous isomorphism?

Compactness and connectedness are preserved by homeomorphism, in the sense that if two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are homeomorphic and $(X,d_X)$ is compact then it follows that $(Y,d_Y)$ is ...
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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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### 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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### Uniform spaces, Dugundji chapter 9, section 11

I think I'm encountering a lot of errors in the section of Dugundji about uniform spaces. Let me write out my concerns. We now use a uniformity to derive a topology. Any uniformity $\mathscr{F}$ in \$...
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