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Questions tagged [uniform-spaces]

In the mathematical field of Topology, a uniform space is a generalization of the concept of metric spaces in which, unlike what happens in general topological spaces, it is possible to compare neighborhoods of distinct points. In uniform spaces, it is possible to define the concepts of uniform continuity, uniform convergence, and of complete space (as in the metric spaces, but unlike what happens in topological spaces in general).

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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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Proof of the “second half of the Heine-Cantor theorem”

A pseudometric on a set $X$ is a metric except it need not be Hausdorff. A gauge $\mathcal D_X$ on $X$ is a an ideal of pseudometrics, i.e.: $\mathcal D_X \neq \emptyset$ $d_1,d_2 \in \mathcal D_X \...
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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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totally disconnected compact Hausdorff uniform space

Let $(X, \mathcal{U})$ be a compact and Hausdorff uniform space. For $D\in\mathcal{U}$, the sequence $\{x_i\}_{i\in\mathbb{Z}}$ is called a $D$- chain if $(x_i, x_{i+1})\in D$ for all $i\in\mathbb{Z}$...
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102 views

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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$D$ 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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What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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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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Completeness of Continuous Functions on Uniform Spaces

So I'm trying to find the most general setting in which I can talk about completeness of function spaces. In metric spaces, it's simple to show that the space $C(X,Y)$ of continuous functions $X\...
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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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quasi-uniformisation of bitopological spaces

By a well-known result of Pervin every topological space is quasi-uniformisable. Since a quasi-uniformity always induces two topologies, one naturally obtains from a quasi-uniform space a ...
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Convergent net which is not Cauchy

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...
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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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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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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 explain ...
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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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Uniquely Promoting a TVS over the Rationals to a TVS over the Reals

I have a Topological Vector Space (TVS) ($I, \oplus , \otimes $) over $ \mathbb Q$ and I want to uniquely extend its scalar multiplication to $ \mathbb R $, so that it is promoted to a TVS over $ \...
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Is there a version of closing lemma in the case of uniform space?

Let $(X, \mathcal{U})$ be a uniform space and $f:X\to X$ be a homeomorphism. For symmetric element $A\in\mathcal{U}$, let $A(f)$ be the set of all homeomorphisms $g:X\to X$ such that $(g(x), f(x))\in ...
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A property of neighborhoods of the diagonal $\Delta\subset Y\times Y$ obtained from a paracompact space $Y$

Let $Y$ be a topological space and $V\subset Y\times Y$. For each $y\in Y$, we define $$V[y]=\{z\in Y\,:\,(y,z)\in V\}.$$ It is possible to show that, if $U\subset Y\times Y$ is an open set and $\...
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proof of theorem $7$ of Some further results on ideal convergence in topological spaces Pratulananda Das

this paper theorem $7.$ Theorem $7$. Let $(X,U)$ be a boundedly compact uniform space. Then the mapping $θ_I :(bs(X),U˜ ) → (K(X),U_H )$ is uniformly continuous. Proof: Let $U ∈ U$. Choose a $V ∈ U$ ...
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Theorem 20.4 in Munkres' TOPOLOGY, 2nd edition Generalized: the product, uniform, and box topologies on arbitrary products

Let $\left\{ \ \left( X_\alpha, d_\alpha \right) \ \colon \ \alpha \in J \ \right\}$ be a collection of metric spaces, where $J$ is some (non-empty) set of indices. Let $$X \colon= \prod_{\alpha \in J}...
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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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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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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 boundedness 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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Proof verification: totally bounded iff every filter has a cauchy superfilter

It would be appreciated if someone could check if these proofs are correct (in Schechter's HAF, the 'only if' direction of the theorem below is essentially left as an exercise with a hint, so I wanted ...
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Why are uniform spaces important?

Why are uniform spaces important? I've thought of two possible good answers: Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and ...
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Hausdorff completion of uniform spaces

I'm working on constructing a completion of uniform spaces. As far as possible I'd like to follow the construction in the case of metric spaces that uses equivalence classes of Cauchy sequences ...
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Question about totally disconnected compact uniform space

Let $(X, \mathcal{U})$ be a compact uniform space. For $D\in\mathcal{U}$, the sequence $\{x_i\}_{i\in\mathbb{Z}}$ is called a $D$- chain if $(x_i, x_{i+1})\in D$ for all $i\in\mathbb{Z}$. Let $X$ be ...
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Non-complete uniform space

If we consider a Cauchy sequence which is not convergent in a non complete metric space we see that the sequence wants to converge a point but it can not find its limit in the space. The divergence of ...
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Is there any kind of distance measurement metric that does not satisfy the triangle principle?

Firstly I am not sure if the definition of "the triangle principle" is used correctly. What I actually want to express is, in a 3D space, if point $\mathbf{X}$ is an interior point of some other ...
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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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For entourage $D\in \mathcal{U}$, is there $x_n\in X$ with $\operatorname{Fix}(f)=\bigcup_{n\in\mathbb{N}}D[x_n]\cap\operatorname{Fix}(f)$?

Let $(X, \mathcal{U})$ be a strongly Lindeloff uniform space ( it is complete separable) and $f:X\to X$ be a homeomorphism. A point $x\in X$ is called a fixed point of $f$ if $f(x)=x$. Denote by $Fix(...
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$\Delta_X$ is $\mu$- isolated set

Let $g:X\times X\to X\times X$ be a continuous function and $(X, \mathcal{U})$ be a compact uniform space. Also for entourage $D\in\mathcal{U}$ suppose \begin{equation} \{z\in X\times X: g^n(z)\in D,...
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Is there a proper set in a locally compact, paracompact uniform space?

Let $X$ be a uniform space with uniformity $\mathcal{U}$ and natural topology of it is locally compact and paracompact space. (the natural topology, $\tau_{\mathcal{U}}$, on $X$ is the family of all ...
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Is there a $D$- chain between two point of a connected component in an uniform space?

A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map of the space into a discrete space is a constant map also a topological space is connected if and only if ...
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Let $\mathcal{U}_\pi$ be quotient uniform with respect $\pi:(X,\mathcal{U})\to Y$

It is known that given a uniform space $(X, \mathcal{U})$ and a function $\pi$ from $X$ onto an arbitrary set $Y$, it is possible to define a uniformity on $Y$, in the following way: \begin{equation*} ...
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Invertibility of Cellular Automata over $A^G$ for locally finite G.

This is an exercise from "Cellular Automata and Groups" by T. Ceccherini-Silberstein and M. Coornaert. Let $G$ be a locally finite group and let $A$ be an arbitrary set. Let $\tau: A^G \rightarrow A^...
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countable set of open sets in a compact uniform space

Let $(X, \mathcal{U})$ be a compact uniform space which is not metrizable and and $\{U_i\}_{i=0}^{\infty}$ be a countable set of $U_i\in \mathcal{U}$ with $U_{i-1}\subseteq U_i$ and $U_1\subseteq V$. ...
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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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Is a minimal Cauchy filter bounded?

Let $E$ be a $K$-vector space and $A, B$ two subsets of $E$. We say $A$ absorbs $B$ if there is a $\alpha>0$ such that $B \subseteq \lambda A$ for all $\lambda\in K$ such that $\lambda \geq ...
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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 $a,...
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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 $$\...