Questions tagged [metric-spaces]
Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
16,092 questions
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Metric space is a topological space
Wanna make sure that when we are using (X, d), it represents a metric space, which is also a topological space. It is expressed in a different way, since d is a map not a collection of sets, but we ...
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What metric spaces have the property that, if we go nearer to two endpoints, we also go nearer to inner points?
Let $(X,d)$ be a geodesic metric space (informally, a space that contains a "shortest path" between any two points). Let $A, C$ be two points in $X$, and let $B$ be any point on a geodesic ...
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Convex sets in general metric spaces
A set of points in a Euclidean space is called convex if it contains every line segment between two points in the set. An equivalent definition is:
a set $C$ is convex if for every two points $x,y \...
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If the distortion of a compact set $X\subset \mathbb{R}^n$ is less than $\frac{\pi}{2}$ then $\pi_1(X)=0$.
I am currently reading M. Gromov's "Metric Structures for Riemannian and Non Riemannian Spaces" and I am aiming to study Gromov-Hausdorff convergence, Asymptotic geometry etc. I am stuck in ...
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Two types difinition of the distance function
Let $\Omega \subset \mathbb{R}^n$ be a domain, $n \ge 2$. I'm wondering that for any
$x \in \Omega$, do we have
$$
\text{dist}(x,\partial \Omega) = \text{dist}(x, \Omega^c)?
$$
It seems natural. ...
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Arzela-Ascoli lemma for separable (not necessarily compact) space: Can the subsequence converge to non-continuous function?
In Royden's Real Analysis there is an Arzela-Ascoli Lemma for separable space (which does not need compactness):
$X$ is a separable metric space and $(f_n)$ is an equicontinuous sequence of real ...
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Finding the boundary of $\{(x, \sin(\frac{1}{x})) : x > 0\} \subset \mathbb{R}^2$.
The following problem was given as an exercise for my course on Intro to Metric Spaces.
Question :-
Define a set $U := \{(x, \sin(\frac{1}{x})) : x > 0\} \subset \mathbb{R}^2$. Find out the ...
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No boundary wave function, quantum cosmology and metrics [closed]
I've been reading up on the No Boundary wave function and how to compute saddle point approximation integrals that do not converge using Picard Lefschetz theory. I see how the calculations work - but ...
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Is it wrong to say that the distance locally approximates the metric in a manifold?
I am doing some data analysis where I use the Riemannian distances between matrices in a manifold (Symmetric Positive Definite matrices).
In my application, the formula for the Riemannian metric ...
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missing step in proof of open subspace of E is E
Let $X$ be un vector subspace of a metric vector space $(E,d)$.
A very classical result is that $X$ open $\Leftrightarrow X=E$.
The proof relies on:
$\vec{0}\in X$
$X$ open $\Rightarrow\exists r\in \...
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The L.U.B of a Set that is Bounded Above but has No Greatest Element is a Cluster Point
I want to prove that if $S$ is a nonempty set of $\mathbb{R}$ that is bounded from above but has no greatest element, then l.u.b. $S$ is a cluster point of S.
I know a direct proof might be simpler, ...
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Exercise 2.6 in Katok's book "Fuchsian Groups"
Exercise 2.6 in Svetlana Katok's book Fuchsian Groups asserts the following
Let $G$ be a subgroup of isometries of a proper metric space $X$. Then the following are equivalent:
(i)/(ii) The action is ...
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Little help in Rudin' s Functional Analysis 3.22
If $0<p<1$, show that $l^p$ contains a compact set $K$ whose convex hull is unbounded.
Attempt
Consider the sequence $x_n$ defined by $x_n(n) = n^{p-1}$ and $x_n(m) = 0$ if $m \neq n$ and the ...
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Is the subset closed iff it is complete? [closed]
Saying that the range of a linear map is closed is the same as saying that its range is a complete set? I'm struggling precisely with the term complete set in this context.
Thanks in advance.
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Existence of transitive set of a homeomorphism
Let $X$ be a compact Hausdorff space without isolated points and $f:X\rightarrow X$ be a homeomorphism such that the $\mathbb{Z}$-action on $X$ that is defined by $f$ is minimal. Namely, for each $x\...
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Let $\ell_2=\{(x_n): \sum_{n=1}^{\infty}|x_n|^2\lt \infty\}$ and $A=\{(x_n):|x_n|\leq \frac 1n,n\in \Bbb N\}\subseteq\ell_2.$ Show that $A$ is closed.
Let $\ell_2=\{(x_n)\subseteq \Bbb C: \sum_{n=1}^{\infty}|x_n|^2\lt \infty\}$ and $A=\{(x_n):|x_n|\leq \frac 1n,n\in \Bbb N\}\subseteq X=\ell_2.$ Show that $A$ is closed.
My solution goes like this:
...
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Cusp-singularity in metric on the plane
Consider polar coordinates $r,\theta$ with $0 \leq r, 0 \leq \theta \leq 2\pi$ on the 2D plane. The flat metric is given by $$\mathrm{d}s^2 = \mathrm{d}r^2 + a(r)^2 \mathrm{d} \theta^2$$ with $$a(r) = ...
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Proof of a result concerning limits of subsequences
I attempted to prove a result related to limits of subsequences so here is the theorem:
Theorem: Let $\lim_{n\to\infty} a_n=L$ and $a_{n_k}$ be a subsequence of $a_n$. Then $\lim_{k\to\infty} a_{n_k}=...
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Clarification required: No non empty metric space is open in every possible superspace
I am reading Michael Searcoid-"Metric Spaces", where, in page 64, example 4.6.1, the following is shown:
Theorem: There exists no non-empty metric space which is open relative to every ...
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Questions about S. Solecki, Analytic ideals and their applications.
I read from "S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99 (1999),51–71."
In the first; I can't understand what is the analytic ideal? I know what the ideal ...
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Continuous functions wrt railway metric
On $\mathbb{R}^n$ we have the french railway metric:
$$ d(x,y) = \cases{\lVert x-y \rVert \text{ , if } x,y,0 \text{ are collinear} \\ \lVert x \rVert + \lVert y \rVert \text{ , else}} $$
This ...
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Determine whether the set $A=\{(0,0)\neq (x,y)\in\mathbb R^2\mid x^2+y^2\leq |x|\}$ is connected.
This is an exercise from my Intro to Real Analysis class. The original questions asks to determine whether $A$ is closed, compact, and connected. I believe I was able to demonstrate closedness and ...
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For any two points x,y in compact metric space $(M,\rho)$ we have z in M : $\rho(x,z)=\rho(x,y)/2$, show there is u in M : $\rho(x,u)=\rho(x,y)/3$
Here is my attempt at a solution. Could anyone advise about the validity or points of improvement?
The core of the proof relies on generating a sequence of midpoints given the two starting points $x$ ...
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Determine whether the set $\{A\in M_{2\times3}(\mathbb R) \mid \operatorname{rank}(A)<2\}$ is connected.
This exercise question from my Intro to Real Analysis class asks me to prove/disprove that the set of rank-deficient, real $2\times 3$ matrices is connected. I've managed to prove the set is closed, ...
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How to prove a subspace of a normed space is closed?
Let $(X,||.||)$ be Banach space. Let $Y=S(\vec0,1)$ be a subspace of $(X,||.||)$.
I want to show that $Y$ is closed in $(X,||.||)$, so that I can say $Y$ is complete, because we have a theorem "A ...
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Gromov hyperbolicity is a quasi-isometric invariant for roughly geodesic metric spaces
Let $(X,d)$ be a metric space. We say a continuous path $\gamma:[a,b]\to X$ is a rough geodesic if there exists $C>0$ such that for any $x,y\in [a,b]$, $|x-y|-C\leq d(\gamma(x),\gamma(y))\leq |x-y|+...
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Calculations of generalized Stokes' Theorem on Riemannian manifold
I'm starting to work on some problems which require the evaluations of scalar functions residing on very smooth manifolds. We want to push the calculations onto the boundaries using Stokes', however I'...
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How to compute the Hausdorff distance between two circles in $\mathbb R^2$
I’m working on computing the Hausdorff distance between two circles in $\mathbb R^2$ and need help with the specific calculation. The equations of the circles are given as:
$x^2+y^2=4$ (a circle with ...
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Specific codomain of a metric
Given $(X,d)$ a metric space and $a \in [0,\infty)$. Is it possible to have to codomain of $d$ be the set $[0,a) \cup (a,\infty)$? In other words, can we find a metric on some space such that the ...
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Why is Hawaiian Earring Sequentially Compact?
I am not looking for a really formal proof, but I remember the reasoning was based on the fact that a subsequence can be found on one of the circles. However I cannot justify this and I don't have any ...
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Difference of a regulr open and another open set in $\mathbb{R}^2$ must have non empty interior if it has positive Lebesgue measure?
$A \subset \mathbb{R}^2$, open, bounded and convex with non empty interior.
$B \subsetneq A$, also open, s.t. $A \setminus B$ has positive Lebesgue measure. Must $A \setminus B$ have a non empty ...
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What is the smallest regular open set containing a given open set in $\mathbb{R}^2$?
Suppose $A \subsetneq \mathbb{R}^2$ is a bounded open set. We know it's regular if
$$
A=\operatorname{int}(\bar{A}) .
$$
Suppose $A$ is not regular.
By assumption, of course, $A=\operatorname{int}(A) ....
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If $A \times B$ is closed in $(X \times Y, d)$, show that $A$ is closed in $X$ and $B$ is closed in $Y$, where A and B are nonempty subsets of X and Y
I am studying some basic topology online, and saw a question:
Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Define
$$
d((x_{1}, y_{1}), (x_{2}, y_{2})) = \max\{ d_{X}(x_{1}, x_{2}), d_{Y}(y_{1},...
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Generalized Dini's Theorem
Dini's Theorem says: let $f_n:K\to\mathbb{R}$ such that
(a) $K$ is compact
(b) $f_{n+1}(x)\le f_n(x)$ for all $n$ and $x\in K$
(c) $f_n$ is continuous and there is a continuous $f:K\to\mathbb{R}$ for ...
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Unnecessary assumptions in Proposition 3.28 of Bridson and Haefliger's book Metric Spaces of Non-positive Curvature
In Bridson and Haefliger's book Metric Spaces of Non-positive Curvature we are given the following proposition (Proposition 3.28)
Let $p:\tilde{X}\rightarrow X$ be a map of length spaces such that
(1) ...
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Prove that a subset $E$ of a metric space $X$ is open iff there is a continuous $f\colon X\to\mathbb{R}$ such that $E=\{x\in X\mid f(x)\gt0\}$ [duplicate]
Prove that a subset $E$ of a metric space $X$ is open if and only if there exists a continuous function $f\colon X\to \mathbb{R}$ such that $E=\{x\in X\mid f(x)\gt 0\}$.
I would appreciate guidance ...
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Characterization of completeness by total-boundedness.
Some time ago our Functional Analysis professor stated (without proof) a theorem characterizing the completeness of metric spaces where one of the conditions was precisely 'complete and totally ...
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Prove that $L_0([0,1]) $ is a topological vector space which is not locally convex and whose convergence is the convergence in measure
Let $L_0([0,1]) = \{ f : [0,1] \to \mathbb{R} : f \text{ is measurable} \}$. We define
$$
d(f, g) = \int_0^1 \frac{|f(x) - g(x)|}{1 + |f(x) - g(x)|} \, dx.
$$
I am interested in proving that:
$d$ ...
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Some doubts regarding topological vector spaces
I am currently studying topological vector spaces and I would like to clear up some doubts.
We know that, in a topological vector space $X$, $U$ is a neighborhood of zero iff $U + a$ is a ...
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78
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X metrizable iff homeomorphic to a closed subspace of $[0,1]^{w}$
I am studying for an exam and in a recommended exercise I have to show the following equivalence:
$X$ is a compact Hausdorff space with a countable basis iff $X$ is homeomorphic to a closed subspace ...
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Continuity of the induced map of fixed points of a product map: Banach Contraction Principle
Let $X$ be a complete metric space, $Y$ a topological space. $f:X\times Y\to X$ a continuous map such that each $f_y:X\to X$ mapping $x\mapsto f(x,y)$ is a contraction, that is, there exists a $0<\...
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Give an example of a weak contraction on a complete metric space that is not a contraction
I was given the following question: Give an example of a complete metric space $(X, d)$ and $f:X \rightarrow X$ such that $d(f(x), f(y)) < d(x, y)$ for all distinct $x, y \in X$, but f is not a ...
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81
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The intersection of all open sets containing $x$ is $\{x\}\$
I was given this question: If $(X, d)$ is a metric space, then the intersection of all open sets sets containing a point $x \in X$ is $\{x\}$.
I gave the following solution: Suppose for a ...
- 395
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48
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Application of Nash's Embedding Theorem for Constructing Pseudo-Riemannian Metrics
As you know, it is possible to define a Riemannian metric on a $\sigma$-compact manifold.
The proof using Nash's Whitney embedding theorem proceeds as follows:
Let $M$ be a $\sigma$-compact manifold.
...
- 53
1
vote
1
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41
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Non-metrizable quotient despite a closed equivalence relation on a separable metric space
Nik Weaver has the following exercise in Chapter 1 of his Measure Theory and Functional Analysis:
1.21. Find an example of a closed equivalence relation on a separable metric
space such that the ...
- 4,540
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$d_1,d_2 $ are equivalent metrics $\iff$ they have the same convergent sequences
I read that: If $d_1,d_2 $ are metrics on $X \neq \emptyset $ then
$d_1,d_2 $ are equivalent metrics $\iff$ they have the same convergent sequences.
I have proved $(\Rightarrow). $
Also I can prove $(\...
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53
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Dual of a metric space: $\operatorname{Lip}_0(X)$
Let $(X,d)$ be a metric space, and let $0$ be any point of $X$, we define $\operatorname{Lip}_0(X)=\{f: X \to \mathbb{R}: \text{f is lispschitz and } f(0)=0\}$.
We know that there is an isometric ...
user1073077
2
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53
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Projection into metric spaces
Let $(X,d)$ be a metric space and $M\subseteq X$. What could be a projection $X$ onto $M$? Is there any way to define this generally?
I saw that it can be defined like this: The projection
$P: X \to M$...
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How to graph balls in metric spaces?
Metric space is given as: $d: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$
$$d((x_1,y_1), (x_2,y_2)) = \left\{\begin{matrix}
|x_1-x_2|+|y_1-y_2|; & \exists k\in \mathbb{R} \text{such that} (...
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Is it true that for any $\varepsilon$ exists some compact K that $P(K) > 1 - \varepsilon$
Is it true that if $ \Omega $ is a complete metric space, $F$ is its Borel $ \sigma $ -algebra, then for every positive $ \varepsilon $ there exists a compact $ K $ such that $ \mathbb{P}(K) > 1 - \...
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