Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Is the intersection of a closed set and a compact set always compact?

I am going through Rudin's Principles of Mathematical Analysis in preparation for the masters exam, and I am seeking clarification on a corollary. Theorem 2.34 states that compact sets in metric ...
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0answers
163 views

l1-metric and cut metric equivalence

I would like to show that the following two statements are equivalent. Let (A, d) be an n-point metric space. And B set of $\binom{n}{2}$ pairs of points of A. $\exists t \geq 1$, integer m, and ...
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924 views

Showing that a set X with a trivial topology is not metrizable

This exercise is from a "challenge" list that my analysis teacher gave us to do: Let $X$ be a set. There are two trivial topologies: Indiscrete (not sure if this is the actual name, as I am ...
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1answer
362 views

Extension of Plancherel theorem to $\ell^2$

Can Plancherel's theorem, which was originally defined for $L^2$ spaces (rather, functions in $L^1\cap L^2$) be extended to $\ell^2$ spaces? How would one do that, or is it very obvious/intuitive? If ...
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Do results from any $L^p$ space for functions hold in the equivalent $\ell^p$ spaces for infinite sequences?

For e.g., is $\ell^2$ self-dual like $L^2$? If some $x[n]\in\ell^1\cap\ell^2$, then does it have a Fourier transform in $\ell^2$?
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1answer
575 views

Metrizable weak topology of closed ball in $c_0$

It is well known that if $X$ is a Banach space and $X^*$ contains separating sequence (i.e. sequence $(f_n) \subset X^{*}$ such that $f_n x = 0$, $n = 1,2, \dots$, implies that $x=0$) and $K \subset ...
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1answer
190 views

Continuity of identity map across isometric metric spaces

Show that $F: (\mathbb{R}^n, \varepsilon) \rightarrow (\mathbb{R}^n, \rho), \quad \mathbf{x} \mapsto \mathbf{x}$ is continuous, where $\varepsilon$ is the Euclidean metric and ...
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1answer
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If x is a limit point of a set S, then every open ball centered at x contains infinitely many points of S?

I'm told that the following statement is true: "If a limit point of the set S is defined as x, then every open ball that is centered at x contains infinitely many points of S." Yet I can't begin to ...
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1answer
391 views

Proving that two metric spaces are isometric

I think (hope) I'm on the right track with this problem, but there are details that I can't seem to work out. I've also struggled to find examples of this sort of problem to assist me. Let $\rho ...
2
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2answers
659 views

A subset $G$ of $\mathbb{R}^n$ is open iff the complement of $G$ is closed

So I'm covering material for my upcoming final exam, and I have a sneaking suspicion that my teacher will ask us to prove the following theorem: A subset $G$ of $\mathbb{R}^n$ is open iff the ...
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228 views

Calculating the Epsilon Neighborhood of line segments in 3d

I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an ...
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1answer
69 views

Proving continuity of a mapping of a space onto the unit sphere?

It seems obvious that in an arbitrary normed space $(X, \|\cdot\|)$ a mapping $T$ defined as $$ T(x) = \begin{cases} \frac{x}{\|x\|} & t \neq 0 \\\\ 0 & t = 0 \end{cases} $$ ...
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1answer
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Triangle inequality for positive definite symmetric real matrix

Show that \begin{equation} \rho(\mathbf{x},\mathbf{y}) = \sqrt{\sum\limits_{i,j=1}^n a_{ij} (x_i-y_i)(x_j-y_j)}, \end{equation} with $a_{ij} = a_{ji}$, $\mathbf{x} = (x_1,\ldots \;, x_n)$ and ...
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1answer
289 views

Continuity of a function from the Cartesian product of a metric space

OK, so, given that $(X, \rho)$ is a metric space, endow $\mathbb{R}$ with the ordinary Euclidean metric $\varepsilon$ and $X \times X$ by $(\rho \times \rho) \left((x_1,y_1),(x_2,y_2)\right) = ...
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193 views

Isoperimetric inequalities of a group

How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being ...
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4answers
316 views

Is it possible to prove that the metric space is an open set without choice?

Suppose that $(X, \rho)$ is a metric space, $|X| > 1$. Is it possible to prove that $X$ is an open set without assuming the axiom of choice? As I understand it, the challenge is to find a way to ...
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701 views

Subset of a bounded set is bounded

Show that if a set $A$ in a metric space is bounded, so is each subset $B \subseteq A$.
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155 views

How to prove that the sequence of $x_n = (1,\frac{1}{2}, \frac{1}{3}, … \frac{1}{n}, 0, 0…)$ does not converge under $\|\cdot\|_1$?

I'm reviewing past assignments and am still having trouble formulating a proof for this: Consider the sequence $(x_n)$, where $x_n = (1,\frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, 0, 0, \ldots)$. ...
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Examples of non symmetric distances

It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of ...
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Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
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Difference between limit point and limit

What is the difference between limit point of a sequence and limit of a sequence. Can it be unique?
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5answers
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What is the difference between topological and metric spaces?

What is the difference between a topological and a metric space?
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How to prove that convergence is equivalent to pointwise convergence in $C[0,1]$ with the integral norm?

I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of ...
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1answer
162 views

Largest number of equidistant points

Given a metric space, we can check what the largest number of equidistant points are (ie, such that the distance between any two of these points is the same). Of course, this might not be finite, as ...
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427 views

The Class of Non-empty Compact Subsets of a Compact Metric Space is Compact

This is a question from my homework for a real analysis course. Please hint only. Let $M$ be a compact metric space. Let $\mathbb{K}$ be the class of non-empty compact subsets of $M$. The ...
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2answers
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Proof of Triangle Inequality on $(\mathbb{R}^n, d_p)$

I have to prove the triangle inequality $(|x_1 - z_1|^p + |x_2 - z_2|^p)^{1/p} \leq (|x_1 - y_1|^p + |x_2 - y_2|^p)^{1/p} + (|y_1 - z_1|^p + |y_2 - z_2|^p)^{1/p}$ for $p \geq 1$ on $\mathbb{R}^2$. ...
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1answer
304 views

$A,B \subset (X,d)$ and $A$ is open dense subset, $B$ is dense then is $A \cap B$ dense?

I am trying to solve this problem, and i think i did something, but finally i couldn't get the conclusion. The question is: Let $(X,d)$ be a metric space and let $A,B \subset X$. If $A$ is an open ...
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Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?

Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space? Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
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Given a metric function between a set of abstract points, what is the best way to plot them on a 2D space?

I have a list of several entities, all of each have a numerical relationship to each other that defines an abstract distance. Is there a mathematical way to plot all of these on a 2D space, turning ...
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Projection to space with smaller dimension that saves a distance

Suppose we have a countable set of objects $\{x_i|i \in [1..m]\}$ in a metric space $(\mathbb R^n,d_n)$ and a map ($F$) mapping the objects to objects in a metric space $(\mathbb R^1,d_1)$. For each ...
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A map on the unit ball

Let $B$ be the unit ball of $\ell^2(\mathbb{N})$, i.e. $B=\lbrace x\in \ell^2(\mathbb{N}): \|x\|\le 1\rbrace.$ For each $x=(x_1,x_2,\cdots)\in B$, let $$f(x)=(1-\|x\|,x_1,x_2,\cdots).$$ Define ...
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Isometry in compact metric spaces

Why is the following true? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
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1answer
738 views

Unit sphere compactness in a metric space

Define $A$ as a nonempty set, $\mathcal{B}:=\{f: A \rightarrow \mathbb{R}: f(A) \text{is bounded} \} ,d_\infty:=\text{sup}\{|f(x)-g(x)|:x \in A\}$. For which $A$ is $\overline {B_1(0)} \subset ...
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2answers
610 views

Closure of a subset in a metric space

Let $(X,d)$ be a metric space and $S \subset X$. Show that $d_S(x):=\text{inf}\{d(x,s): s \in S\}=0 \Leftrightarrow x \in \overline S .$ Notes: $\overline S$ is the closure of S. Maybe you can use ...
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244 views

Boundary in metric spaces

I should prove or give a counterexample for: $\partial (\bigcup_{i \in \mathbb{N}} A_i) \subset \bigcup_{i \in \mathbb{N}} (\partial A_i)$ Where $\partial$ is the boundary and (X,d) is a metric ...
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4answers
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Unit circle metric

Let $S^1$ the unit circle in $\mathbb{R}^2$ and $$d: S^1\times S^1\to\mathbb{R}$$ $$d(\theta_1,\theta_2) = \left\{ \begin{array}{ll} |\theta_1-\theta_2| & \mbox{if } ...
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Non separable locally compact connected metric space

do you have an example of a non separable locally compact connected metric space? Thank you
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1answer
148 views

If no Cauchy subsequence exists, must a uniformly separated subsequence exist?

Given a sequence $(x_n)$ in a metric space $M$, call it uniformly separated if all pairwise distances $d(x_n,x_m)$ between distinct terms are uniformly bounded away from zero. Suppose that a given ...
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3answers
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Finding the fixed points of a contraction

Banach's fixed point theorem gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction. I'm interested in how to calculate the ...
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4answers
832 views

Developing the unit circle in geometries with different metrics: beyond taxi cabs

My class had a good time redeveloping the unit circle under the taxicab metric. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" ...
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Verifying some basics of the metric topology

While reading about general topology, I started looking at the metric topology. At one point, the interior of a topological space $X$ is defined as such: $${E}^{\circ}=\{p\in E\ | \ ...
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Two metrics induce the same topology, but one is complete and the other isn't

I'm looking for an example of two metrics that induce the same topology, but so that one metric is complete and the other is not (Since it is known that completeness isn't a topological invariant). ...
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Gradient flows in metric spaces

What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe ...
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Which metric spaces are totally bounded?

A subset $S$ of a metric space $X$ is totally bounded if for any $r>0$, $S$ can be covered by a finite number of $X$-balls of radius $r$. A metric space $X$ is totally bounded if it is a totally ...
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Motivation behind the definition of complete metric space

What is motivation behind the definition of a complete metric space? Intuitively,a complete metric is complete if they are no points missing from it. How does the definition of completeness (in ...
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Difference between complete and closed set

What is the difference between a complete metric space and a closed set? Can a set be closed but not complete?
3
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1answer
287 views

On two different proofs

This question is mainly to understand the meaning of my professor's correction to a proof of a theorem I gave during an oral examination. The question was to show that $\text{diam}A = ...
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1answer
248 views

A question about open covers

Let $\mathcal{F}$ be an open cover of a metric space $(X,d)$. I have a question concerning some algebraic aspects of $\mathcal{F}$ and $X$. First of all, is it always possible to consider $(X,d)$ a ...
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Notions of equivalent metrics

Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...