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14
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1answer
255 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 ...
11
votes
2answers
288 views

Topological groups are completely regular

I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all ...
11
votes
1answer
204 views

Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
11
votes
1answer
141 views

a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
10
votes
1answer
243 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 ...
7
votes
1answer
230 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, ...
6
votes
1answer
271 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?
6
votes
1answer
90 views

Separation axioms in uniform spaces

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ ...
6
votes
1answer
359 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 ...
5
votes
1answer
281 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 ...
5
votes
1answer
77 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 ...
5
votes
1answer
454 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 $\overline{D}$.
4
votes
1answer
210 views

Is a quotient of a complete group always complete?

Let $\: \langle G,\cdot,\mathcal{T}\hspace{0.01 in} \rangle \:$ be a $\big($$\text{T}_0$$\big)$ topological group. $\;\;$ Let $H$ be a closed normal subgroup of $G$. Set $\;\; \mathbf{G} \: = \: ...
4
votes
1answer
68 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?
4
votes
1answer
156 views

Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?

AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
4
votes
1answer
62 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
132 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
133 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.
3
votes
2answers
78 views

“Uniform groups” (similar to topological groups)?

Why have I heard about topological groups, but nothing about "uniform groups" (uniform spaces endowed with a group)?
3
votes
2answers
244 views

Spaces with the property: Uniformly continuous equals continuous

I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces. This fact inspires me to ask: Are there ...
3
votes
2answers
209 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 ...
3
votes
2answers
146 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 ...
3
votes
1answer
117 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
27 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 $M$ ...
3
votes
1answer
48 views

lemma about uniform topology

It's my first post on this site, so please forgive me for all mistakes I've made during writing it. I can't understand the proof of this lemma: Let $( X,\mathcal U )$ be an uniform space, $x\in X$ ...
3
votes
1answer
505 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 ...
3
votes
1answer
174 views

Benefits from using the uniform structure of compact Hausdorff spaces

It is well known that every compact Hausdorff space admits a unique (necessarily complete) uniform structure which is compatible with the topology, and every continuous function from such a space to a ...
3
votes
0answers
48 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
0answers
63 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 ...
2
votes
2answers
132 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 ...
2
votes
1answer
85 views

What are interesting properties of totally bounded uniform spaces?

I work on reloids, a generalization of uniform spaces. As such I am interested about properties of totally bounded uniform spaces. My question: What are interesting properties of totally bounded ...
2
votes
1answer
107 views

Completion of a linear order that is a dense subspace of a compact space.

Suppose $D$ is a linearly ordered space which is densely embedded in a compact Hausdorff space $K$. What can we say about the relation between $K$ and $\overline D$, the completion of $D$. Is one a ...
2
votes
2answers
88 views

continuous images of Cauchy sequences in topological groups

on page 102 of Atiyah and MacDonald's "Introduction to Commutative Algebra", they state that if $G$ and $H$ are abelian topological groups and $f$ is a continuous homomorphism from $G$ to $H$, then ...
2
votes
1answer
54 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
42 views

$C$-embedding in uniform spaces

Every Hausdorff uniform space $X$ has a Hausdorff completion $C_X$. Is it true that $X$ is $C$-embedded in $C_X$? How about the completion with respect to its finest uniformity $\mu_X$?
2
votes
1answer
77 views

Equivalent definitions of complete uniform space

A uniform space $(X,\mathcal{U})$ is called complete if every Cauchy filter converges. In this page, Brian M.Scott says that, In a uniform space every Cauchy filter converges iff every Cauchy ...
2
votes
0answers
26 views

A totally bounded uniformity and certain filters

Conjecture For every totally bounded uniform space $(U;F)$ and filters $\mathcal{X}$, $\mathcal{Y}$ on $U$ such that $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$ ...
2
votes
0answers
120 views

hyperspace of a complete uniform space need not be complete

I want to know the counter example for: Hyperspace of an arbitrary complete uniform space need not be complete. The hyperspace of a uniform space $(X,\mathscr D)$ is obtained by forming the set ...
2
votes
0answers
73 views

Weaker Than The Weak Topology?

The weak topology on a Banach Space $E$ is defined to have sub-base consisting of open balls of the form $B_\alpha(x,r) = \lbrace y \in E \colon \vert \alpha(x-y) \vert < r\rbrace $ for each $x \in ...
2
votes
1answer
64 views

Uniformity is generated by pseudometrics

How to prove that every uniform space is generated by a family of pseudometric spaces? You may offer me a book. In Engelking this theorem is presented without a proof. In Willard it is a exercise. ...
2
votes
0answers
40 views

Terminology for some special sets

The following predicates are used in the definition of total boundness of uniform spaces: a space is bounded if the predicate holds for every entourage, specifically for the first predicate (the ...
2
votes
0answers
165 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 ...
1
vote
1answer
183 views

A uniformly continuous function between totally bounded uniform spaces

Let $X$ and $Y$ is a uniform spaces. Let $f$ is a uniformly continuous surjective function $X\rightarrow Y$. Conjecture: If $X$ is totally bounded then $Y$ is also totally bounded.
1
vote
1answer
77 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 ...
1
vote
1answer
29 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
1answer
53 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
75 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
17 views

Different uniform spaces having the same set of Cauchy filters

I want to understand how Cauchy space is different than uniform space. For this I need an example: An example of two different uniform spaces having for both of them the same set of Cauchy filters?
1
vote
1answer
74 views

Neighborhood of diagonal

Consider a uniform space $X$ (with induced topology). What of the following can be a subset of the other? Which of the following is always a subset of the other? Neighborhood of the diagonal in the ...
1
vote
1answer
58 views

Question on Cauchy filters

Let $(X,\mathcal{V})$ is a uniform space and $\xi$ is a Cauchy filter on $(X,\mathcal{V})$. $o(\xi)$ is the family of all open subsets of $X$ containing at least one element of $\xi$. what ...