In the mathematical field of topology, a uniform space is a set with a uniform structure. (Def: http://en.m.wikipedia.org/wiki/Uniform_space)

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products of uniform space

$X$ is a topological space(or even a nonempty set), and $(Y,Φ)$ is a uniform space, then $Y^X$ is a uniform space, too. $\widetilde{U}=\{(f,g):$for any $x\in X, (f(x),g(x))\in U\}$, ...
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A uniform space must be a symmetric space.

The topology of a uniformizable space is always a symmetric topology; that is, the space is an $R_{0}$-space. How to prove it? It must be a simple question, but I can't write it down. $X$ is an ...
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Comparing the Samuel and Stone-Čech compactifications of a Hausdorff topological group

Let $G$ be an Hausdorff topological group and let $\beta G$ be the Stone-Čech compactification of $G$. Now, $G$ is also a uniform space with respect to the so-called right uniformity. Let $S(G)$ be ...
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A new proof of Tychonoff's theorem from the subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) : Theorem: Let $(X,\mathcal{U})$ be a ...
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Total order and its order topology

I noticed that the natural order of the Reals alone, being complete ( satisfying LUB ) , is able to prove that the induced order topology is complete ( every cauchy sequence converges ). We are ...
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In the category of uniform spaces, is the completion of a quotient map also a quotient map?

Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous map $f: X \to Y$ is a quotient map if for every map $g$ from $Y$ to a uniform space $Z$ such that $g \circ f$ is ...
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Question regarding uniform spaces and equicontinuity number 2

Following the already answered question: Question regarding uniform spaces and equicontinuity in the context of proposition 27. How do we know that indeed every element in the p-closure of G is a ...
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Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
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Are continuous functions on uniform spaces Cauchy continuous?

Suppose $X,Y$ are uniform spaces. Since uniform structures give topologies $X,Y$ are naturally topological spaces, so we can consider continuous functions $X \to Y$. What I am wondering is if $f ...
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When does pointwise convergence on compact space imply uniform convergence?

I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then ...
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Analog of an open map for uniform structures

In topology, a function is continuous if the inverse image of an open set is open. A function is open if the image of an open set is open. Uniformity continuity can be defined in a similar way as ...
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non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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Question regarding equicontinuity and joint continuity

I can't seem to prove this: Let G be an equicontinuous subset of the continuous functions between E and F. Then the open point topology on G is jointly continuous. For the definition of ...
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Compactification: density of a uniform space $X$ in the spectrum of $UC^b(X)$

First, a small motivation: Suppose we are looking for a compactification of uniform spaces, satisfying an universal property similar to the one of the Stone-Čech compactification of a locally compact ...
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Questions about a topological category

Given topological spaces $(X_i,\tau_i)$ with sets $\mathbf S_i=\{\mathcal A\in \tau_i^2|\mathcal A\supseteq\Delta X_i^2\}$, where $\tau^2$ is the product topology and $\Delta$ is the diagonal. The ...
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Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta ...
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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 ...
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$D$ 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 ...
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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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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?
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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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TVS: 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 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 $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$ ...
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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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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?
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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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The neighbourhood filter of each point is a minimal Cauchy filter

Wikipedia says (for every fixed uniform space) "The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter." Please help me to prove ...
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Maximal Cauchy filter

Product of two filters $\mathcal{A}$ and $\mathcal{B}$ is defined as the filter $\mathcal{A}\times\mathcal{B}$ generated by the base $$\{A\times B \,|\, A\in\mathcal{A}, A\in\mathcal{B} \}.$$ I call ...
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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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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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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, $x\in X$ ...
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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 ...
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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 ...
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“Uniform groups” (similar to topological groups)?

Why have I heard about topological groups, but nothing about "uniform groups" (uniform spaces endowed with a group)?
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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 ...
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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) ...
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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. ...
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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 $V \in \mathcal{U}$ ...
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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 ...
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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 ...
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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 ...
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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 ...