19
votes
A good place to read about uniform spaces
For a general audience it can be interesting to know that uniform spaces are just one of two opposite generalizations of metric spaces. Measuring distances with the help of metric, we can be ...
- 42k
17
votes
What is the structure preserved by strong equivalence of metrics?
Metrics are strongly equivalent if the identity mapping $Id:(X,d_1)\to (X,d_2)$ is bi-Lipschitz. They preserve the class of Lipschitz mappings.
Roughly speaking classical topology deals with notions ...
- 28k
12
votes
Accepted
Totally bounded spaces and axiom of choice
The issue here is that a metric space might not have non-trivial (read: not eventually constant) Cauchy sequences. For example, if the underlying space is a Dedekind finite set.
Indeed it is ...
- 39.8k
8
votes
Using the Stone-Weierstrass theorem to solve an integral limit
Since $f$ is increasing, $\mu(0,x)=f(x)$ defines a (here: continuous) measure on $[0,c]$, which we can also view as a measure on $\mathbb R$ with support in this set. By Fubini, the RHS equals
$$
\...
- 24.2k
8
votes
Uniform spaces as condensed sets
I think there are a few things to untangle here.
First, as concerns your highlighted question, it seems that you've answered it yourself: outside the compact Hausdorff case (where the uniform ...
- 9,283
8
votes
What is the structure preserved by strong equivalence of metrics?
First of all, there are two aspects to Lipschitz maps between metric spaces: Local (having implications such as differentiability properties, etc) and global. The two aspects have different non-metric ...
- 12.3k
7
votes
Accepted
Open mapping theorem for complete non-metrizable spaces?
Question 1. Such results have been studied in detail—-a good reference is Köthe‘s monograph on topological linear spaces. You could also look up the concept of webbed spaces (de Wilde).
For question ...
- 2,472
7
votes
Accepted
Convergent net in a quasi-uniform space which is not Cauchy
There are several definitions of Cauchy filters on a quasi-uniform space $(X,\mathcal U)$ [K]. For instance, a filter $\mathcal F$ on $(X,\mathcal U)$ is called
a left $K$-Cauchy (resp. right $K$-...
- 5,409
5
votes
Using the Stone-Weierstrass theorem to solve an integral limit
Consider first $g$ continuous, strictly increasing in $[0,1]$ with $g(0)=0$, $g(1)=a$
$$ \frac {1}{t}\int_0^1 f(x)(g(x+t)-g(x))\, dx=\frac{1}{t} \int_0^1 f(x) dx \int_{g(x)}^{g(x+t)} ds=\frac{1}{t}\...
- 6,035
4
votes
What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?
As a first step you may want to work problem 8.5.16 in Engelking's General Topology. Keep in mind that a compact Hausdorff space has a unique uniformity, the sets of all neighbourhoods of the diagonal,...
- 11.4k
4
votes
Accepted
Does each $\omega$-narrow topological group have countable discrete cellularity?
Mikhail Tkachenko informed me that the problem has a counterexample, constructed in Example 8.2.1 of his book with Arhangelskii.
This example looks as follows. Consider the uncountable power $C_2^{\...
- 42k
4
votes
Accepted
Even covers and collectionwise normal spaces
As implied by Tyrone, the claim that a fully normal space is strongly collectionwise normal is true. See, for example, the result in H. J. Cohen's paper Sur un problème de M. Dieudonné (translated ...
- 716
3
votes
Accepted
A a question about the metrization of uniform spaces
The relationship between the two pseudometrics is the following: $\rho(x,y)=2\cdot d(x,y)$ for all $x$ and $y$.
We compare the definitions of the pseudometrics in both books.
Both start with a ...
- 11.4k
3
votes
Using the Stone-Weierstrass theorem to solve an integral limit
I understand the aim is an elementary proof; here is one.
Let $\omega$ be a modulus of continuity for $f$ on $[0,c]$. Then, for all $0<t\le c-1$
$$\int_0^1|f(x+t)-f(x)|^2dx\le \omega(t)\int_0^1\big(...
- 60.6k
3
votes
Quotient of compact metrizable space in Hausdorff space
For the Cantor starcase function $f:C\to[0,1]$ from the standard ternary Cantor set $C$ onto the interval $[0,1]$ and for the standard Euclidean metric $d$ on $C$ the quotient pseudometric $d_\sim$ is ...
- 42k
3
votes
A good place to read about uniform spaces
About motivation. (Late answer, sorry) I don't know what you have chosen and how it went, but had I have found your question in time, I would have recommended pseudometrics. Your audience knows metric ...
- 3,738
3
votes
Accepted
Cartesian powers of uniform spaces
E.g. $C(\mathbb{R} ,[0,1])$ in this function space uniformity is metrisable in this uniformity, using the $\sup$-metric $d(f,g) = \sup \{|f(x) - g(x)|: x \in \mathbb{R}\}$. While the product $\mathbb{...
- 5,407
2
votes
Duality between large and small scale structures
You might try taking a look at the following:
https://trace.tennessee.edu/utk_graddiss/3705/
It is a dissertation from 2016 that uses "large scale spaces" instead of coarse spaces. They are ...
- 213
2
votes
Accepted
Uniformly Converging Metrization of Uniform Structure
$f_k$ converges uniformly for $1\leq\beta\leq 2.$
I will argue that for any $\epsilon>0$ and any path $z_0,z_1,\dots,z_{k-1},z_k,$ there is a sub-path $z_0,z_{i_1},\dots,z_{i_{m-1}},z_k$ such that ...
- 1,338
2
votes
Accepted
Extensions of bounded uniformly continuous functions
$\DeclareMathOperator{\R}{\mathbb R}
\DeclareMathOperator{\eps}{\varepsilon}$
If you prefer to define uniformities in terms of a family $D$ of pseudometrics you can reduce the theorem to pseudometric ...
- 16.5k
2
votes
Open mapping theorem for complete non-metrizable spaces?
Another example for question 2 is the following: There are linear partial differential operators with constant coefficients (e.g., the wave operator) and open sets $\Omega \subseteq \mathbb R^3$ (e.g.,...
- 16.5k
2
votes
Uniform spaces as condensed sets
Here is an essentially tautological answer. The notion of uniformity makes sense also for condensed sets -- it is a condensed set $X$ together with certain subsets $U\subset X\times X$ termed ...
- 21.3k
1
vote
Accepted
A question about uniformities generated by pseudometrics
One direction:
let $\varepsilon>0$ and taken $N$ such that $\sum_{n>N}a_n<\frac12\varepsilon$. Take $\delta>0$ such that $\delta\cdot\sum_{n\le N}a_n<\frac12\varepsilon$. Now if $d_n(x,...
- 11.4k
1
vote
Open mapping theorem for complete non-metrizable spaces?
In fact, we have a partial opposite for general topological vector spaces:
(1) Theorem. Let $E$ be a vector space, and let $\mathcal T_1 \subseteq \mathcal T_2$ be linear topologies on $E$ such ...
1
vote
Open mapping theorem for complete non-metrizable spaces?
$\def\sp{\kern.3mm}\def\TT{{\mathscr T}}$Still further (counter)examples of the situation in Q2 are obtained from Propositions 4.4.3 (p. 81) and 6.6.7 (p. 111) and Example 6.10.L (p. 123) in Jarchow's ...
- 3,584
1
vote
Open mapping theorem for complete non-metrizable spaces?
A good reference for some of the most general versions of the open mapping theorem (and its cousin the closed graph theorem) is the book
Meise, Vogt: Introduction to Functional analysis, 1997
...
- 2,388
1
vote
Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?
To construct a counterexample, start with any completely regular topological space $T$ for which there are unbounded continuous functions (that is, the space is not pseudocompact). For example, $T$ ...
- 1,074
1
vote
Accepted
When is the unitary dual of a lscs group uniformizable?
A topological space $X$ is uniformizable iff it is completely regular. Glimm's Theorem states that a second countable group is Type I iff $\widehat{G}$ is $T_0$. Since a $T_0$ uniformizable space is ...
- 3,816
1
vote
Topology generated by complete and incomplete uniformities
$]0,1[$ and the real line are, famously, homeomorphic but the latter is complete whereas the former is not (both under the usual metric).
- 88
Only top scored, non community-wiki answers of a minimum length are eligible