# Questions tagged [uniform-spaces]

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46
questions

**5**

votes

**5**answers

303 views

### Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...

**2**

votes

**1**answer

135 views

### Does each $\omega$-narrow topological group have countable discrete cellularity?

A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable.
A family $\mathcal F$ of subsets of a topological space ...

**2**

votes

**1**answer

82 views

### Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...

**28**

votes

**3**answers

709 views

### What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...

**4**

votes

**1**answer

120 views

### What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

It is known that if $(X, d)$ is a compact metric space, then hyperspace $2^X= \{A\subseteq X: A\text{ is closed set} \}$ is a compact space with Hausdorff metric
What can say about $2^X= \{A\...

**3**

votes

**0**answers

83 views

### Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...

**2**

votes

**1**answer

163 views

### Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...

**7**

votes

**1**answer

414 views

### Totally bounded spaces and axiom of choice

Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...

**4**

votes

**1**answer

110 views

### When is the unitary dual of a lscs group uniformizable?

Let $G$ be a locally compact, second countable group. We equip the unitary dual $\widehat{G}$ with the Fell topology. I am looking for conditions which guarantee that the topological space $\widehat{G}...

**0**

votes

**1**answer

70 views

### Topology generated by complete and incomplete uniformities [closed]

Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?

**4**

votes

**1**answer

136 views

### Convergent net in a quasi-uniform space which is not Cauchy

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...

**7**

votes

**0**answers

165 views

### Results that are easier in a metric space

Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces?
In particular, I'm ...

**5**

votes

**0**answers

129 views

### Has the Roelcke completion of a topological group any reasonable algebraic structure?

It is well-known that each topological group $G$ carries (at least) four natural uniformities:
the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...

**2**

votes

**1**answer

80 views

### Cartesian powers of uniform spaces

In the nlab entry on uniform spaces they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped ...

**2**

votes

**1**answer

100 views

### Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity.
To construct Fine uniformities, Let ...