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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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 ...
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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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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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meaning of the notation $\mathcal{U}/V$

If $\mathcal{U}$ is an open cover of a uniform space and $\mathcal{V}$ is a uniform cover then what is the meaning of the notation $\mathcal{U}/V$ where $V \in \mathcal{V}$?
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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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How to geometrically interpret the composition of entourages?

In my topology course, we have been working for quite some time now with uniform spaces. I understand the metric background and I can work abstractly with uniformities, so that is great. However, it ...
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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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“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 ...
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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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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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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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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 ...
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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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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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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 ...
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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 ...
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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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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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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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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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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 ...
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Pointfree generalization of uniform spaces?

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

Are completeness and being totally bounded somehow related with each other for uniform spaces?
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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 ...
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Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. To construct Fine uniformities, Let ...
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Explicit formulas for meets and joins of uniform spaces

I want explicit formulas for meets and joins (and finite meets and joins) for sets of uniform spaces (where uniformities are ordered by inclusion). And also for proximity spaces. I am also ...
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Prove that $x^2$ maps Cauchy-filters to Cauchy-filters

Let $(\mathbb{R},\mathcal{U})$ be a uniform space where $\mathcal{U}$ is the filter on $\mathbb{R}^2$ generated by the basis $\{S_\varepsilon \colon \varepsilon > 0\}$ with $$S_\varepsilon := ...
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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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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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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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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 ...