# Tagged Questions

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### 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 ...
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### 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 $x$.
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### Uniform Continuity $\implies$ Continuity

In metric spaces it is a well known fact that uniformly continuous functions are indeed continuous at any point. What about uniform spaces? How can I prove this? (with the definition of topology in ...
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### Topological Vector Space: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
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### What makes metric spaces special?

This is not a question about what is special about a metric space in itself; instead, I'm wondering what sets metric spaces apart from uniform spaces? An explanation is in order. As a parallel, when ...
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### The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite ...
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### 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 ...
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### 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? ...
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### Cauchy filters defined for proximity spaces?

I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula: $$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$ ...
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### 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 $\forall A\in S:A\times A\subseteq E$. ...
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### 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 ...
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### Completeness and being totally bounded

Are completeness and being totally bounded somehow related with each other for uniform spaces?
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### 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 ...
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### 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 $X'$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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, $x\in X$ ...
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### 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?
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### 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 ...
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