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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Is a sequentially compact (non-metrizable) uniform space totally bounded?

First some topological definitions in terms of nets and sequences: A topological space $(X, \tau)$ is compact iff every net has a convergent subnet sequentially compact iff every sequence has a ...
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Criterion for relative compactness in uniform spaces

I am having problems in understanding a criterion for relative compactness given in a book (see below for details if you are interested) on SPDEs. However, I think it just invokes a pretty general ...
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Existence of an unique uniform space, generated by a family of coverings.

It is known, that given a family $M$ of subsets of a set $X$, there exists a unique, minimal topology $\tau$, such that $M \subset \tau$, we get this topology as the intersection of all topologies ...
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Generates a covering uniformity on a set X, by an entourage uniformity.

A family $\mu$ of coverings of a set $X$ is called a covering uniformity if it satisfies the following conditions: If $\frak U,\frak V\in\mu$ there is a $\frak W\in\mu$ with $\frak W\prec ...
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The definition of two-sided uniformity

In the field of uniform spaces, what does it mean to be a two-sided uniformity? Could not find a clear definition of this online.
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Unit group over a real closed field

Let $F$ be a real closed field. Let ${\mathbb{S}^1}(F)$ denote the abelian group $(S^1(F),*)$ where $S^1(F) = \{(a,b) \in F^2 \ | \ a^2 + b^2 = 1\}$ with the law $(a,b)*(c,d) = (ac - bd,ad + bc)$. ...
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The separated uniform space associated with $(X,\mathfrak{U})$

If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in ...
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How can we construct *Fine uniformities*?

Given a uniformizable (w.r.t. entourage uniformity) space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. A uniform ...
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Is the long line completely uniformizable?

The long line $L$ is uniformizable; in fact, as $L$ is a locally compact Hausdorff space we can explicitly write down a uniformity for it: If $\hat{L}$ is the one-point compactification of $L$, then ...
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Are the sections of entourages in a uniform space open?

Wikipedia's article on uniform spaces defines the following. A nonempty family $\mathcal{U}$ of subsets $U \subseteq X \times X$ is a uniform structure if it satisfies the following axioms: ...
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do the uniformly continuous functions to the reals determine the uniformity?

It is well known that the completely regular spaces $X$ are characterized as those topological spaces whose topology is recovered from $C(X)$, the set of continuous functions $X\to \mathbb R$. In ...
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Boundedness in uniform spaces?

After looking a bit at uniform spaces, as their general definition seems relevant to the study of topological vector spaces, it seems that they provide just enough structure to define the notion of ...
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equality in uniform space and topological groups

I wanted to ask the following: If I have a topological group $G$, I know I can create a base for a uniform space as follows: for each $U$ a neighborhood of e, we define $V_u= \{(x,y):x^{-1}y\in U\}$. ...
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precompactness and boundedness in uniform space

Consider a uniform space $(X,\mathscr{U})$. For an entourage $U\in\mathscr{U}$, one says that a set $M$ is small of order $U$, if $M\times M\subseteq U$. $P\subset X$ is precompact if for every ...
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closure of bounded set in uniform space

I have stumbled upon the following statement but I could not see why it is true. Let $(X,V)$ be a uniform space and $A$ bounded in it. Bounded means that for each $W\in V $ there is a finite set ...
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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 $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta ...
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$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 uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
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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 ...