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13
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1answer
178 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 ...
4
votes
1answer
196 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 ...
2
votes
1answer
72 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. ...
1
vote
1answer
12 views

$D$ a closed entourage, $K$ compact subset, show that $D[K]$ is closed.

I'm studying for my topology exam and have come across a question that I can't solve. To state the problem more clearly: For $D$ a closed entourage in a uniform space $X$, and $K$ a compact subset of ...
1
vote
1answer
88 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 ...
0
votes
1answer
28 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
0
votes
1answer
27 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
1answer
39 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
52 views

Two definitions of totally bounded uniform spaces

Wikipedia gives this definition of totally bounded uniform space: a subset $S$ of a uniform space $X$ is totally bounded if and only if, given any entourage $E$ in $X$, there exists a finite cover of ...
3
votes
0answers
50 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
65 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
0answers
134 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
93 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
0answers
41 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
174 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
0answers
68 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 $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$ ...
1
vote
0answers
36 views

Completeness and being totally bounded

Are completeness and being totally bounded somehow related with each other for uniform spaces?
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?
1
vote
0answers
72 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 ...
0
votes
0answers
19 views

Question related to Uniform Space

I have questions related to Uniform Space; If $X$ is a countable discrete space, then how to show that finest pre compact uniformity on $X$ admits a countable base of entourages. If $\mho$ is a ...
0
votes
0answers
32 views

Bounded Point in Uniform Spaces

I'm currently studying uniform spaces and have come across a problem I don't know how to solve. Given any vicinity $U$ of a non-discrete uniform space, I want to prove that for every pair of points ...
0
votes
0answers
14 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
0
votes
0answers
39 views

A relation between certain families of filters and filters on a cartesian product of sets

(By filters, I will mean all filters on a set, including the improper filter.) The product $\mathcal{A}\times\mathcal{B}$ of two filters is the filter defined by the base $\{ A\times B \,|\, ...
0
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0answers
82 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
27 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 ...
0
votes
0answers
25 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
0answers
36 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 ...