0
votes
1answer
22 views

A stronger concept than total boundness

A space, every proper principal filter of which is refined by a Cauchy filter, is called totally bounded. Is there a term (and theory) about a stronger concept: a space every proper filter of which ...
0
votes
0answers
71 views

Totally bounded uniform spaces vs proximity spaces (need proof)

nLab says "The category of totally bounded uniform spaces and uniformly continuous functions is equivalent to the category of proximity spaces and proximally continuous functions". How to prove this? ...
0
votes
0answers
21 views

Cauchy filters defined for proximity spaces?

I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula: $$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$ ...
0
votes
1answer
82 views

Three definition of total boundness (for uniform spaces)

The following are three definitions of a totally bounded uniform spaces on a set $U$: For every entourage $E$ there exists a finite cover $S$ of $U$ such that $\forall A\in S:A\times A\subseteq E$. ...
0
votes
0answers
19 views

Minimal Cauchy filters in Cauchy spaces

It is well known that "every Cauchy filter contains a unique minimal Cauchy filter" (Wikipedia) for both metric spaces and uniform spaces (see also this question and answer). Does this theorem ...
1
vote
0answers
33 views

Completeness and being totally bounded

Are completeness and being totally bounded somehow related with each other for uniform spaces?
0
votes
0answers
22 views

Topology generated by a uniformity

Let $\mathfrak{X}$ be the complete lattice of all filters (including the improper filter) on $U\times U$ (for some set $U$), with the order being the set inclusion of the filters. Consider the ...
0
votes
1answer
146 views

Bourbaki on the fact that continuous function on a compact is uniformly continuous

I am now looking theorem 2 in paragraph 4.1 of: Bourbaki. "Elements of Mathematics General Topology. Part 1". THEOREM 2. Every continuous mapping $f$ of a compact space $X$ into a uniform space $X'$ ...
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 ...
0
votes
0answers
26 views

Closure operation in a uniform topology.

The closure operation in the uniform topology is $$\overline{A} = \bigcap_{E \in \mathcal{U}} E \cdot A$$ where $E \cdot A$ is the set of all $E$-left-relatives of $A$. I cannot seem to prove that ...
0
votes
1answer
25 views

Uniformizable space implies T3 space

A topological space $(X, \mathscr{T})$ is uniformizable if there exists a uniform structure on $X$ that induces $\mathscr{T}$. I am trying to prove that every uniformizable space on $X$ is T3. To do ...
1
vote
1answer
53 views

Weight of a uniformity and topological cardinal invariants

A family $\mathcal B\subset\mathcal U$ is called a base for the uniformity $\mathcal U$ if for every $V\in\mathcal U$ there exists a $W\in\mathcal B$ such that $W\subset V$. The smallest cardinal ...
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, $x\in X$ ...
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 ...
0
votes
2answers
76 views

Can I Have Some More Examples Of Uniform Spaces Please?

I have been reading about uniform spaces and topological groups. There does not look to be a lot of literature on the topic, much less accesible literature, and the books that I have been reading do ...
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)?
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 ...
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. ...
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 $V \in \mathcal{U}$ ...
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
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 ...
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
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 $\mu_X$?
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 ...
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 ...
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 ...
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
82 views

“Principal uniform space” vs “discrete uniform space”?

Which terms are better for a uniform space such that the set of entourages is a principal filter? "Principal uniform space" or "discrete uniform space"? "Principal uniformity" or "discrete ...
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
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 ...
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 ...
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?
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
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 ...
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?
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
2answers
59 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
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 ...
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 ...
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 $M$ ...
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 ...
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 $\overline{D}$.
0
votes
1answer
62 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
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?
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 ...
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 ...
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.
1
vote
0answers
60 views

Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?