Tagged Questions

3
votes
0answers
25 views

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 ...
3
votes
1answer
75 views

Dieudonné complete and topologically complete are equivalent for every space $X$.

How can we show that: For every topological space $X$ the following conditions are equivalent: A space $X$ is topologically complete if $X$ is homeomorphic to a closed subspace of a product ...
5
votes
1answer
192 views

$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
1
vote
0answers
36 views

Is the fine uniformity of a metric space, metric?

Let $d$ be a metric on the set $X$ and let $\mathcal T$ be it's topology and $\mathcal D$ be the finest uniformity on $X$ which induces $\mathcal T$. Does $\mathcal D$ have a countable base?!
4
votes
1answer
56 views

Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
3
votes
1answer
93 views

Is every $T_4$ topological space divisible?

A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the ...
3
votes
1answer
101 views

Is a compact Hausdorff uniform space fine?

Let $\mathcal D_1$ and $\mathcal D_2$ be two uniformities on $X$ which produce the same topologies on $X$ (say $\mathcal T= \mathcal T _{\mathcal D_1}=\mathcal T _{\mathcal D_2}$). If $(X,\mathcal ...
6
votes
0answers
199 views

non-hausdorff completion of a uniform space.

Let $(X,\mathcal U)$ be a Hausdorff uniform space. Can $(X,\mathcal U)$ have a non-hausdorff completion?
1
vote
1answer
27 views

Anti-symmetric property of embedding in topological spaces.

$(X,\mathcal T)$ and $(Y,\mathcal S)$ are topological spaces. $X$ can be embedded homeomorphically in $Y$ and $Y$ can be imbedded homeomorphically in $X$. Are $X$ and $Y$ homeomorphic? How about ...
1
vote
2answers
41 views

Open sets in a topology produced by metrics.

For each $i\in I$, $$d_i:X^2\to \Bbb R$$ is a metric and $\mathcal T$ is the coarsest topology on $X$ containing all topologies produced by the $d_i$ . For $U\subseteq X$ we have: $$\forall x \in ...
3
votes
1answer
74 views

Open sets in uniform and box topology

Let $\mathbb{R}^{\omega}$ denote the product of countably-many copies of $\mathbb{R}$. Let $\bar{d}$ be the following metric on $\mathbb{R}$: $$\bar{d}(x,y)=\inf\{|x-y|,1\}$$ The uniform metric on ...
0
votes
0answers
48 views

Pseudometrics that produce a uniform space.

Let $(X,\mathcal D)$ be a uniform space and let $(D_n)$ be a sequence of symmetric entourages such that: $$D_{n+1}\circ D_{n+1}\circ D_{n+1}\subseteq D_n$$ $$D_1=X^2$$ I define $$d:X^2\to\Bbb R$$ ...
2
votes
2answers
64 views

Definitions and coincidences of the topology of pointwise convergence and the uniformity of uniform convergence

I was wondering how "the topology of pointwise convergence" is defined on $Y^X$ where $X$ is a set and $Y$ is a topological space? Are there more than one topologies that can topologize pointwise ...
3
votes
1answer
22 views

Uniform structure induced by a mapping from its codomain to its domain?

Suppose $X$ is a set and $Y$ is a uniform space with uniform structure $M$. Given a mapping $f: X \to Y$, I was wondering if $f$ can induce a uniform structure on $X$ from the uniform structure $M$ ...
0
votes
0answers
23 views

Every unifom space can be embedded in a product of (pseudo-)metric spaces.

In this book page 18 I found this theorem: Every uniform space can be embedded in a product of metric spaces. I googled, I found only a similar theorem about pseudo-metric spaces. I think it ...
4
votes
1answer
48 views

If $\mathcal{B}$ in $X$ converges to $x$, it accumulates at $x$ and if $X$ is Hausdorff, $x$ is a unique point of accumulation.

Essentially I need feedback, mostly on writing style and accuracy and tightness of arguments. Suppose $\mathcal{B}$ converges to $x$. For every open neighbourhood $U(x)$ of $x$ in $X$ there exists ...
4
votes
1answer
145 views

Is $\overline{D}\subseteq D\circ D$ in a uniform space?

Suppose $(X,\mathcal{D})$ is a uniform space and $D\in\mathcal{D}$. Is it true that $$\overline{D}\subseteq D\circ D,$$? here we use the product topology to define $\overline{D}$.
0
votes
1answer
43 views

Definition of product uniform space.

Let for each $i\in I$, $(X_i,\mathcal{D}_i)$ be a uniform space. How is the product uniform space defined? Does it produce the product (Tychonoff) topology on $\prod_{i\in I}{X_i}$?
1
vote
1answer
40 views

Is the intersection of two uniformities a uniformity?

Is there any two uniformities $\mathcal{D}_1$ and $\mathcal{D}_2$ on a set $X$ such that $$\mathcal{D}_1\cap \mathcal{D}_2$$ is not a uniformity on $X$?
1
vote
1answer
64 views

Suppose every complete subset of a uniform space is closed. Is it Hausdorff?

Suppose $(X,\mathcal{D})$ is a uniform space. for each nonempty subset $A\subseteq X$, if the subspace $(A,\mathcal{D}_A)$ is complete then $A$ is closed. Is $(X,\mathcal{D})$ Hausdorff?
1
vote
1answer
47 views

Counterexample for an Alternate Definition for Uniform Spaces.

Let $\mathcal{D}$ be a filter on $X^2$ such that: $(\forall D\in\mathcal{D})(\Delta(X)\subseteq D)$. $(\forall D\in\mathcal{D})(D\circ D\in \mathcal{D})$. $(\forall D\in\mathcal{D})(D^{-1}\in ...
2
votes
1answer
55 views

a counterexample for Uniform Spaces

Uniform Space is a generalization of metric spaces . In a uniform space the closure of a singleton $\{x\}$ is the intersection of all neighborhoods of $x$. Find an infinite topological space such ...
4
votes
1answer
91 views

Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?

And what else can be said, if so? In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
4
votes
1answer
98 views

Why are locally compact groups Weil complete?

Why are locally compact groups Weil complete? Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent. Thank you, and sorry if I have bad writing.
1
vote
0answers
52 views

Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?
6
votes
1answer
274 views

Compactly supported continuous function is uniformly continuous.

What is the most general space where compactly supported continuous functions are uniformly continuous? I managed to prove this for metric spaces but I am interested if it also holds in more general ...
1
vote
0answers
55 views

Uniform covers and partitions of unity

A cover $\mathcal C$ of a uniform space $(X,\mathcal U)$ is called a uniform cover if there is $U\in \mathcal U$ such that the cover $\{U(x):x\in X\}$ refines $\mathcal C$. Is it true that to every ...
4
votes
1answer
73 views

Showing a uniformity is complete.

I've seen in various textbooks and notes that if $X$ is paracompact, then the collection of all the neighborhoods of the diagonal is a uniformity. I am trying to show that this uniformity is complete ...
13
votes
1answer
225 views

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 ...
8
votes
1answer
208 views

Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It ...
0
votes
1answer
145 views

Definition of product of uniform spaces

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous. But Springer's encyclopedia ...
6
votes
1answer
171 views

When is a uniform space complete

From Wikipedia: a uniform space is called complete if every Cauchy filter converges. I was wondering if the following three are equivalent in a uniform space: every Cauchy filter converges, ...
3
votes
0answers
57 views

Extension of pseudometrics to Hausdorff completion

Let $(X,\mathcal{U})$ be a uniform space with Hausdorff completion $(X',\mathcal{U}')$ (made by the minimal Cauchy filters). Since $X$ is uniform, $\mathcal{U}$ is generated by pseudometrics ...
0
votes
1answer
98 views

Hausdorff completion of a uniform space with pseudometrics.

I'm having trouble constructing the Hausdorff completion of a uniform space $(X,U)$ using pseudometrics. I know that every uniformity on a space $X$ is made by pseudometrics. Here is my idea: Let ...
2
votes
0answers
144 views

On the Compact Uniformization Theorem

I just read through a proof of the Compact Uniformization Theorem, and I follow it up to the very last line. The proof is: Compact Uniformization Threorem. If $X$ is a compact regular space, then the ...