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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64 views

Why is the Gromov-Hausdorff distance a metric?

The Gromov-Hausdorff distance is: $$ d_{GH}(A,B) = \inf_{f,g}d_H(A',B') $$where $f$ and $g$ are isometric embeddings of $A,B$ into some metric space, and their images are $A', B'$. The inf is taken ...
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55 views

Constructing true metrics in infinite dimensional vector spaces?

Is there an example of a true metric defined on a function space? I'd imagine it is some type of integral involving two functions, and it will return a value that obeys the metric axioms, but I have ...
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1answer
36 views

Quadratic form as generalized distance?

In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin: $$x^{T}x = ||{x}||^2$$ which represents the square of the ...
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1answer
18 views

Lipschitz maps on locally compact groups

Suppose $G$ is a locally compact second countable group. This means that there exists a proper (closed bounded sets are compact) left invariant ($d(gx,gy) = d(x,y) \ \forall g,x,y \in G$) metric on ...
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56 views

Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$

Given the metric space $(X,d)$ with $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$, how can I show that $(X,d)$ is complete? I need to prove that any Cauchy sequence converges, so: If $(x_n)$ is a ...
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1answer
35 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
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1answer
65 views

What is wrong with my brute-force approach to proving that $\mathbb R$ as a metric space obeys the triangle inequality?

In a self-study of metric spaces, I'm looking at the very basic exercise of proving that $(\mathbb R, |y-x|)$ is a metric space. The sticking point was the triangle inequality. I did manage to ...
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1answer
41 views

How to show $G$ is a perfect set that contains no rational points?

For $E:=[0,1]$, since $\Bbb Q\cap E$ is enumerable, let it be $\{q_1,q_2,\cdots\}$. If I remove the elements of $V_1:=(q_1-\frac1{10},q_1+\frac1{10})$ from $E$, I obtain a closed (and compact) set ...
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1answer
29 views

Prove the Supremum is attained.

Let $F$ denote denote the set of real valued functions on $[0,1]$ such that, 1) $ \; |f(x)| \leq 1 \; \forall x \; \in [0,1]$ 2) $ \; |f(x)-f(x')| \leq |x-x'| \; \: \forall x,x' \: \in [0,1] $ ...
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2answers
28 views

Proving that a subset endowed with the discrete metric is both open and closed - choice of radius of the ball around a point

My question is related to proving that any subset $D \subset X$, where $(X,d)$ is a metric space with $d$ being the discrete metric, is both open and closed. I've read some suggestions to a ...
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15 views

Suppose $X$ is a metric space. Let $C$ denote the collection of all dense subsets of $X$. Show that $\bigcap C = iso(X)$

Suppose $X$ is a metric space. Let $C$ denote the collection of all dense subsets of $X$. Show that $\bigcap C = \mathrm{iso}(X)$, where $\mathrm{iso}(X)$ refers to the set of all isolated points of ...
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1answer
40 views

Proving that is $A:X \implies Y$ is a linear operator from metric space X to Y is continuous iff it is bounded bounded

The $\implies$ part interests me. The proof given goes like this: Let $A$ be continuous in 0 (because the 0 vector is in every vector space) $B_y(0,r)=\{y \in Y | \| y\|<r \} \implies \exists ...
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2answers
53 views

How do I rate smoothness of discretely sampled data? (Picture!!!)

In the sense that the following curves pictured in order will be rated 98%, 80%, 40%, 5% smooth approximating by eye. My ideas: (1) If the curves all follow some general shape like a polynomial ...
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1answer
62 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
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1answer
36 views

Show that $A\subseteq B\implies A^{\circ} \subseteq B^{\circ}$ in a different way.

Let $A$ and $B$ be subsets of a metric space $(M,d)$. If $A\subseteq B$, then $A^{\circ} \subseteq B^{\circ}$. Proof : Assume that $a\in A^{\circ}$. Then there exists a $r>0$ such that ...
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1answer
16 views

Separability of $l^{p}$ spaces

How can I prove that the space $l^{p}$ equipped with the norm (for $x=(x_{n}) \in l^{p})$: $||x||_{p}=(\displaystyle\sum_{n}|x_{n}|^{p})^{1/p}$ Is a separable space? (i.e. showing that there is a ...
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1answer
62 views

Homeomorphism $\mathbb{R}^{2}\setminus \mathbb Z^2$ to $\mathbb{R}^{2}\setminus \{ (x,y) \ | \ (x-n)^2+(y-m)^2<\frac{1}{10}, n, m \in\mathbb Z \}$

Show that $\mathbb{R}^{2}\setminus \{(x,y)\, |\, x \text{ and } y \text{ integers }\}$ is homeomorphic to the space $\mathbb{R}^{2}\setminus \big\{(x,y) \ | \text{ there are integers } $n$, $m$ ...
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1answer
25 views

Evaluation function is Lipschitz wrt uniform conv metric

In the book on Brownian motion by Schilling and Praetzsch there is following statement: Let $\mathcal{C}_{(0)}:=\{f\in\mathcal{C}[0,\infty):\ f(0)=0\}$ be the space of all continuous functions ...
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40 views

What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
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3answers
61 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
2
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1answer
41 views

Doubling measure and Riesz Potential

I am currently trying to solve some analysis exercises on metric spaces, but I cannot quite tackle on of them. The exercises read as follows: Define the measure ...
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1answer
47 views

Prove that if the closure of each open ball in compact metric space is the closed ball with the same radius, then any ball in this space is connected

I'm having some difficulty with the following problem in general topology: Prove that if the closure of each open ball in compact metric space is the closed ball with the same center and radius, then ...
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35 views

Using CAT(0) inequality

Let $X$ be a CAT(0) space with metric $d$. Let $p,x,y$ three points on $X$, and let $u,v$ be points on geodesic $[p,x]$ and geodesic $[p,y]$ such that $d(p,u)\geq a,d(p,v)\geq a$,where a is some ...
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1answer
59 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
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1answer
50 views

Prove that metric Space $X$ must be complete

If every nest $F$ of non empty closed subsets of a metric spaces $X$ satisfying $\inf\{diam(A)~|~A \in F\}=0$ has singleton intersection,then $X$ must be complete. Attempt: My textbook gives the ...
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2answers
64 views

Basic topology: having trouble understanding a couple of things in Baby Rudin

I'm reading Baby Rudin's chapter 2 concerning Euclidean space. First, I find these two statements very confusing to me, which seemingly contradict: (1). In Example 2.16 there is a statement that ...
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1answer
57 views

Proof of Cantor's Intersection Theorem

I am going through metric spaces by Michael Searcoid. The text proves the Cantor's Intersection theorem as shown in the image below. I understand the proof. However, just one thing, I am a little in ...
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If $\inf\{diam ~(A)~|~A \in F\} =0.$ Show that $\bigcap F = \emptyset$ or $\bigcap F$ is a singleton set.

Suppose $(X,d)$ is a metric space and that $F$ is a nest of non empty subsets of $X$ for which $\inf\{diam ~(A)~|~A \in F\} =0.$ Show that $\bigcap F = \emptyset$ or $\bigcap F$ is a singleton set. ...
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60 views

Prove that $f_A (x) = d({\{x}\}, A)$, is continuous.

Prove that: Let $(X, d)$ be a metric space, and let $A$ be a subset of $X$. The function $f_A\colon X\rightarrow \mathbb{R}$, defined by $f_A (x) = d({\{x}\}, A)$, is continuous. Honestly, I ...
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24 views

Subset of a $F_{\sigma}$ set is $F_{\sigma}$

Suppose that $X$ is a metric space. Is it always true that for any $F_{\sigma}$ set $A$, any subset $B \subset A$, $B$ is $F_{\sigma}$? It seems correct to me but I have no idea how to prove it.
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A nice way to see that $(\ell^\infty,d_\infty)$ is not separable

Let $(X,d)$ a metric space such that $\exists A\subseteq X$ uncountable and $\exists \epsilon\gt0$ such that $\forall x,y \in A,x\neq y\Rightarrow d(x,y)\gt\epsilon$. Prove that $X$ is not a ...
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42 views

Existence of an metric or a topology so that every subset is compact

Let $X$ be a infinite set. Is there a metric on $X$ such that every sub set of $X$ is compact? What about a topology on $X$? I think that if we can answer first question then we can answer the ...
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Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
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1answer
13 views

Completion of R-integrable functions by L-integrable functions

I read that: "There's an analogy between the completion of rational numbers by real numbers and the completion of Riemann integrable functions by Lebesgue integrable functions". Can someone elaborate ...
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38 views

$(M,d)$ is complete iff all closed and countable subspace of $M$ is complete

A metric space $(M,d)$ is complete $\iff$ if every closed and countable subspace $F\subseteq M$ is complete. $\implies)$ For this implication I use a proposition that says: "If $X$ is a complete ...
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1answer
15 views

Maximal subset with finite Assouad-Nagata dimension

Given some space $X$ with non-finite Assouad-Nagata dimension. Is it possible for a subset $Y \subset X$ with finite Assouad-Nagata dimension to exist such that $Y$ is maximal in the sense that if any ...
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2answers
60 views

Pre-images and local homeomorphisms

I want to prove that if $f: M \to N$ is a local homeomorphism, then for all $y \in N$ we have $f^{-1}(\{y\}) \subset M$ closed and discrete. Here's the catch: this is from an exercise sheet from over ...
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1answer
70 views

When is it mathematically correct to take a limit in certain expresions?

So now I managed to put together a couple of proofs, that each of them use a similar procedure in a crucial step, and I am not sure what are the requirements for this step to be true. First example: ...
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$f:X\rightarrow Y$ a homeomorphism, so $X$ is separable iff $Y$ is separable

Let $X$ and $Y$ metric spaces and $f:X\rightarrow Y$ a homeomorphism. Prove that: $X$ is separable iff $Y$ is separable. My thoughts are: f, as it is defined, is surjective so $f(X)=Y$..that is so ...
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Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
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Completeness of subset of metric space.

Let $\mathcal C[a,b]$ be the space of continuous function $f:[a,b]\to \Bbb R$ with supremum metric. Let $l,m$ be fixed real numbers. Prove that the subset of $\mathcal C[a,b]$ consisting of all ...
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1answer
27 views

$G$ is open in $(A,d)$ iff $G = A \cap U$ for some set $U$ that is open in $(M,d)$

I've been working through Real Analysis by Carothers and need some help understanding the proof of this Proposition. In the forward direction ($G$ open in $A \implies G=A \cap U$), the proof goes ...
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1answer
64 views

union of two connect sets in particular case

Let $(X,d)$ is a metric space and $A,B \subset X$ are connected and $A \cap B = \emptyset$ and $A^- \cap B \neq \emptyset$ ($A^-$ is closure of $A$) now prove or disprove that $A\cup B$ is ...
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1answer
59 views

Why this matrix is not a distance matrix?

While working on this topic, I came across the following matrix $$D=\begin{pmatrix} 0&1&1&2\\ 1&0&\sqrt 2&1\\ 1&\sqrt 2&0&1\\ 2&1&1&0 \end{pmatrix}$$ ...
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Global structure of the Gromov-Hausdorff space

EDIT: now crossposted at mathoverflow (http://mathoverflow.net/questions/212364/on-the-global-structure-of-the-gromov-hausdorff-metric-space) This is a purely idle question, which emerged during a ...
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Prove that X has at most three elements.

Let be $X\subset\mathbb{R}^{2}$ such that a euclidean metric induces in $X$ a metric zero-one. Prove that $X$ has at most three elements I do not know how to start... What's mean "such that a ...
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1answer
55 views

Why are convex metric spaces defined this way?

If my understanding is correct, a metric space $(X, d)$ is called convex if for all $x \in X$, and $y \in X - \{x\}$ there exists some $z \in X -\{x,y\}$ such that: $$d(x,z) + d(y,z) = d(x,y)$$ I can ...
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1answer
44 views

Show triangle inequality in the metric.

Let be $d:\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ a function such that $$a)d(x,y)=0\iff x=y$$ $$b)d(x,z)\leq d(x,y)+d(z,y)$$ Prove that $d$ is a metric. First, let be $z=x$, then $$0\leq ...
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2answers
60 views

If $X$ is a separable metric space and $M \subset X$ is a metric subspace then $M$ is separable.

If $X$ is a separable metric space and $M \subset X$ is a metric subspace then $M$ is separable. So i'm having the following doubt. Since $X$ is separable, there exist $\mathcal{D} \subset X$ dense ...
0
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1answer
105 views

Length of unit circle

Let $\it{l} $ be the length of the unit circumference $\{(x,y):||(x,y)||=1\}$ in an arbitrary norm $||\cdot||$ in $\mathbb{R}^2.$ How to prove or disprove the inequalities $\it{l} \ge 6,\, \it{l} \le ...