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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Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
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29 views

Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
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35 views

Reference for convergence properties of the Hausdorff distance

Consider the following properties of the Hausdorff distance in $\mathbb R^n$. Let $\Omega_n \supset \Omega_{n+1} \supset ...$ a sequence of open, convex and bounded sets with ...
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33 views

The metric identification of a pseudometric on $C(\mathbb{I})$

I have a pseudometric $\mu$ on $C(\mathbb{I})$ defined by $$\mu(f, g) = |f(x_0) - g(x_0)|.$$ I then take the metric identification of $(M, \mu)$ and am asked what familiar space this metric ...
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Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
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35 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
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Does a strictly convex and weak metrizable unit sphere of a Banach space imply separability?

I want know If $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does a strictly convex and weak metrizable unit ...
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36 views

Inequality proof using the triangle inequality

I am reading Kreyszig's Intro to Functional Analysis and am a bit stoked with one of the problems (problem 12 in section 1.1, page 9): Problem: Given a metric space $(X, d)$, show, using the ...
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32 views

$p$-adic metric proof

I need to prove this, Let $p$ be an odd number. It is defined the function $v_p:\mathbb{Q}\to \mathbb{Z}$ as $$v_p\left(p^n\frac{a}{b}\right)=n, \hbox{ if } \mathrm{mcd}(a,p)=\mathrm{mcd}(b,p)=1.$$ ...
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30 views

In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
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38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
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19 views

Easier proof of “countable hypocompactness”

I am interested in the following result, which appears as an old qual problem: Let $X$ be a metric space and $\{U_i\}$ a countable open cover. Prove that there exists a countable open refinement ...
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50 views

Gromov-Hausdorff distance between a “Line segment” and a “Zylinder”

I want to prove the following statement: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = \frac{\pi}{2} $ where $d_{GH}$ denotes the Gromov-Hausdorff ...
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21 views

Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
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107 views

How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
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19 views

How are delta distributions defined in metric spaces?

How are delta distributions defined in metric spaces with continuous metric?
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86 views

Has anyone seen this space before? Does it have a name?

See the space below (the set taken as a subspace of the plane). It sort of looks like a comb, but with the wrap-around portion added, and the lower left corner removed. What would be a good name ...
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46 views

Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
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32 views

For any countable $ A$ , $B \subseteq A \implies B \cap B\space' \ne B $

In which kind of metric spaces is the following true For any non-empty countable set $A$ of the metric space , $B \subseteq A \implies B \cap B\space' \ne B $
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32 views

Is this subset of a finite metric space already named?

Given a finite set, $X$, with a metric, $d(x,y)$ defined on it, I am interested in the following subsets: $S_k\subseteq X$ s.t. $\forall x\in X,\exists s\in S_k:d(x,s)\geq k$ Do such constructions ...
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63 views

Proving that there is no norm for the space of real-valued sequences making it a complete metric space.

Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become ...
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35 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
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21 views

Creating a metric from a pseudometric

Given the following definition of a pseudo-metric on the set $X$ : A pseudo-metric on the set $X$ is a map $d:X \times X \to \Bbb R^+$ such that for all $x ,y \text{ and } z \in X :$ (PM1) $x=y ...
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28 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
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25 views

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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63 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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33 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
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31 views

Question about finite subcovers

I'm having problems wrapping my head around the part with $\rho_i$.Here goes: $A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely ...
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39 views

a complete space

Define a set $$X=\left\{f:\mathbb{R}\rightarrow\mathbb{R}|f \mbox{ is n-times continuously differentiable}\right\}$$ equipped with the norm $$||y||=\max_{\begin{subarray}{l} ...
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Relation between positive definite metric and full basis of a given operator

Let's have some linear space with given indefinite metric. How the fact that metric isn't positive definite is connected with the fact that hermitian (due to the definition of hermicity in a given ...
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91 views

Is there a continuous, strictly increasing function $f: [0,\infty)\to [0,\infty)$ with $f(0) = 0$ such that $\tilde d = f\circ d$ is not a metric?

Is there a continuous, strictly increasing function $f \colon [0,\infty)→ [0,\infty)$ with $f(0) = 0$ such that $\tilde d = f\circ d$ is not a metric? You may take $(X,d)$ to be $\mathbb R$ with the ...
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26 views

Regularity of measures proof

A probability measure $\mathbb P$ on a metric space $(S,d)$ is closed regular if $$ \mathbb P(A) = \sup \{ \mathbb P(F) : F \subseteq A, F \text{ - closed} \} \text{*}$$ with $A\in ...
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Do all subsets of metric spaces have boundry points?

I am learning about metric spaces. I failed to find an aunambiguous answer to my question on Google. So this is the right place to ask: Assume (X,d) is a metric space, and $A \subset X$. If I ...
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Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
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38 views

A basic question on converges in distribution

The distance $d(F,G)$ between two distribution functions is the infimum of those $\epsilon > 0$ such that $G(x-\epsilon) - \epsilon \leq F(x) \leq G(x+\epsilon) + \epsilon \quad\forall x $. Now I ...
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Preimage of a sequence of cauchy

Let $X,Y$ metric spaces. If $f:X\to{Y}$ is continuous, and $\{y_n\}$ is a Cauchy sequence in $Y$. Then, my question is $\{f^{-1}(y_n)\}$ is a Cauchy sequence in $X$? I´m sorry, in a second ...
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Balls and their cardinality

Let metric space $X$ and fixed point $a\in X$ be given. I search for not restrictive assumption under what all balls in metric space $X$ with center in $a$ have the cardinality equal the carinality ...
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49 views

Statistics for random permutations

Let $S_n$ be the symmetric group on $n$ elements, let $d$ be the Cayley distance, and let $m$ be a Haar measure on $S_n$. Let $s$ denote a random permutation with respect to $m$, i.e., $s$ is an ...
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166 views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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progression along geodesics

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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Define a subset of a metric space that is both open and closed.

Define a nonempty subset of a metric space that is both open and closed. The real line with the Euclidean metric $d(x,y)=|x-y|$ is open and closed. If you take two real lines, not connected together, ...
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Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
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Complement of set of all condensation point for an uncountable set of reals is at most countable.

Perfect Set: A set $E \subset X$ is said to be perfect if $E$ is closed in the metric space $(X,d)$ and every point of $E$ is a limit point of $E$. Condensation Point : A point $p \in X$ is said to ...
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Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
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is symmetric chi-squared distance “A” metric?

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric? I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ ...
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How does one “separate” the cartesian product properly?

Say, $\delta>0$, $X$ and $Y$ are metric spaces, $(x_0,y_0)\in X \times Y $, and there is some property $P$ such that $$\forall (x,y) \in X \times Y: \ \ \ \ d \Big( (x_0,y_0), (x,y) \Big) < ...
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The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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155 views

How can one prove that mahalanobis distance is a metric?

How can one prove that mahalanobis distance is a metric? How can one show that these four properties of a metric are valid for mahalanobis distance? 1) d(x, y) ≥ 0 (non-negativity, or separation ...
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Proof that compactness can be characterized by closed sets.

If anybody would be willing to check to see if this proof is correct I would really appreciate it. Prove that a metric space $(X,d)$ is compact if and only if for any family $(C_i)_{i \in I}$ of ...
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space metris disjoint

Let F be a closed subset of a metric space M and p∈M∖F . Show that there are two disjoint open sets G and H in M such that p∈G and H⊆F . I solved well but I think is not the way ... We take a ...