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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define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
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Quasi-isometry of Schreier coset graphs

Let $G$ be a finitely generated group and $H$ a subgroup. For any choice of a generating set $X$ we can form the Schreier coset graph. Is this graph independent of the generating set up to ...
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1answer
23 views

basic question of topology involving compactness and convexity

Consider in $R^n$ a compact and convex set $A$ with $int(A) $ nonempty. then $\overline{int(A)} = A$ ?. i have no idea to prove this. In this direction i only know the following (and hard to ...
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2answers
16 views

Is a closure a disjoint union of limit points and isolated points

Definition) A point $x\in X$ is a limit point of S if every ball $B(x;r)$ contains infinitely many points from $S$. A point $x\in X$ is called an isolated point of S if $\exists r > 0$ such that ...
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1answer
14 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
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2answers
31 views

Product topology and standard euclidean topology over $\mathbb{R}^n$ are equivalent

I would like to know why the product topology and the standard euclidean topology over $\mathbb{R}^n$ are equivalent. I already found the question here: Showing that the product and metric topology ...
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2answers
849 views

What's the motivation behind metric spaces?

So a metric space is a collection of points together with operations, and where we can determine the distance between any of these points. And it must satisfy 4 axioms which are: For all x in that ...
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0answers
11 views

Which Similarity Metric should I use?

I have currently created three similarity metrics, but the point is I have no idea which one I should use and when one is better for use than the other. I am currently using these metrics for people ...
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1answer
35 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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14 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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1answer
163 views

Triangle Inequality on a different normed space

Let $x=(x_1,x_2)$, the norm is given by $[x]=\sqrt{x_1^2+x_1x_2+x_2^2}$ I need to show the triangle inequality holds. So $y=(y_1,y_2)$ and from $[x+y]\le[x]+[y]$ I got ...
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2answers
19 views

What does the sup function mean in the context of metrics for probability measures/distances/differences?

I was studying different probability metrics and distances and came across the following source: ...
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1answer
50 views

Using the topology of uniform convergence for functions over non-compact spaces

Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is ...
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1answer
48 views

Is the following statement true?

Let $(X_1, d_1),\ldots,(X_n, d_n )$ be metric spaces, $ X: = X_1 \times \cdots\times X_n$ be their Cartesian product with metric $d$. Let $ \pi_i : X \to X_i$ be the projection for $ 1 \le i \le n$. ...
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0answers
37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
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1answer
63 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
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1answer
18 views

continuous functions on metric space

Assume $f:X\rightarrow Y$, where $X$ and $Y$ are two metric spaces. If $f(\overline{E})\subset \overline{f(E)}, \, \forall E\subset X$, then how can we prove that $f$ is continuous? Thank you for ...
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1answer
22 views

A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
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1answer
28 views

A simple question on Hausdorff distance

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$. Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then $$ A_n ...
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0answers
12 views

Verify global Lipschitz condition

Consider the realized observations $z_1,...z_i,...,z_n$ i.i.d. Let $$ \hat{Q}_n(\theta)=\frac{1}{n} \sum_{i=1}^{n} 1\{z_i\in S(\theta)\} $$ I have to verify the following global Lipschitz condition: ...
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1answer
27 views

The Open Set $X-\lbrace x \rbrace$

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set I kind of have an idea but I am unsure about it and how to express it. I was thinking about using the ...
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1answer
37 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
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1answer
34 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
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1answer
31 views

How to show that $(C^0((a,b)), d_\infty)$ is not a metric space

Let $d_\infty:C^0([a,b]) \times C^0([a,b]) \to [0,\infty)$ be defined as $$ d_\infty(f,g)=\sup\limits_{x \in [a,b]} \left\{ |f(x) - g(x)| \right\} $$ I have already shown that $(C^0([a,b]), ...
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1answer
28 views

Show that the interior of the set A is empty?

Consider $A = \{(x, \sin\frac{1}{x}) \mid 0< x \leq 1 \}$, a subset of $\mathbb R^2$. Find int($A$). We can see graphically that the interior of $A$ is definitely empty, but I want to check by the ...
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1answer
28 views

Prove that metric space is complete

I have metric space: $$ X = <[0,+\infty), \rho>, \rho(x,y) = |ln(1+x) - ln(1+y)|$$ I know it is complete, but I don't know how to prove it. How can I prove that fact?
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1answer
44 views

Set theory: not really understanding what the question is asking..

Here is the problem. Let $M$ be the metric space of all real numbers, and let $x_0 \in M$. Prove that there exist exactly two isometries of $M$ that leave $x_0$ fixed. I am having trouble ...
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1answer
37 views

Proving the distance function $|f - g|_u = \sup \{ |f(x) - g(x)|: x \in S \}$ defines a metric space

Let $S$ be a closed and bounded subset of $\mathbb{R}$. Define the "functional distance" between $f$ and $g$, both functions from $S$ to $\mathbb{R}$, to be $$ |f - g|_u = \sup \{ |f(x) - g(x)|: x ...
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23 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
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1answer
30 views

Are these metrics complete?

Determine if these subsets of R are complete with the Euclidean metric? a) $[0,\infty)$ b) $(0,\infty)$ I know the definitions of completeness and I know the Euclidean metric, but don't know how to ...
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1answer
30 views

Prove that if $B(x,r)$ and $B(x',r')$ are disjoint $\Longleftrightarrow d(x,x') \ge r+r'$

Assuming that $d(x,x') \ge r+r'$, and proving that they are disjoint is easy. It's the other side that I'm having difficulty with. This seems like a really easy problem, but i'm having difficulty ...
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3answers
43 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
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3answers
45 views

Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
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4answers
45 views

(Proof Checking) Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$.

Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$. I am tempted to use the following argument. Let $U = \{U_i|i\in I\}$ be some open cover of $C\cap ...
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1answer
24 views

find a sequence of closed connected subsets $V_n$ of $\mathbb R^2$ s.t. $V_n\supseteq V_{n+1}$ and $\cap^{\infty}_{i=1}V_i$ is not connected

Find an example of a sequence $V_n$ of closed and connected subsets of the Euclidean plane satisfying $V_n \supseteq V_{n+1}$ such that $\cap_{i=1}^{\infty}V_i$ is not connected. Normally I would ...
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1answer
15 views

Hausdorff distance for empty sets?

The Hausdorff distance is defined for non-empty sets. What would be a reasonable generalization of the definition for the case when one of the sets is empty, if the generalized distance should remain ...
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2answers
27 views

Closure of a particular set, $c_{00}$.

I have been learning the concept of closure in metric spaces and all has made sense so far. However, I have come across a particular example that is troubling me in terms of extracting the possible ...
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1answer
9 views

Closure of interior proper subset of interior of closure

$\newcommand{\intr}{\mathrm{int}}$ I need to give an example of a metric space $(X,d)$ and $A ⊆ X$ so that $\overline{\intr(A)} ⊂ \intr (\overline{A})$, where $\overline{B}$ refers to the closure of ...
2
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1answer
30 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
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0answers
59 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
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1answer
27 views

proving the equivalence of to metrics

Any hints as to how I can prove that $(\mathbb{R}^n,d_\infty)$ and $(\mathbb{R}^n,d_T)$ are topologically equivalent. Where $$d_\infty = \sup\{|x_i-y_i|, |x_2-y_2|,..,|x_i-y_i|\} $$ and $$d_T = ...
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1answer
41 views

Proving a basis exists

How would I show that there exists some set of open balls with rational radius and rational centre such that they are a subset of the reals.That is, $\exists p,q\in \mathbb{Q} $ and $ r,x \in ...
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4answers
881 views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ...
3
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1answer
45 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
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1answer
19 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
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3answers
53 views

proving topological equivalence

How would I show that $$d_E = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$ and $$d_\infty= \sum_{i=1}^n|x_i-y_i|$$ and $$\sup\{|x_1-y_1|,|x_2-y_2|,...,|x_n-y_n|\}$$ are topologically equivalent on $\mathbb{R}^n$? ...
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2answers
72 views

A metric space is compact if and only if its complete and totally bounded.

I just need help with one direction. In particular suppose that $(X,d)$ is complete and totally bounded. Suppose that $X$ is infinite. Take any infinite subset $Y$ in $X$ and then just show that $Y$ ...
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2answers
45 views

proving a metric

I'm trying to show that given a metric $d(x,y)$ show that $d_0(x,y) =\frac{d(x,y)}{1+d(x,y)}$ is also a metric.. It's trivial to show the first two properties, that is, $d_0\geq 0 $ & for $x=y ...
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1answer
55 views

Hausdorff distance and intersection

The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set. Suppose there ...
0
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0answers
20 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...