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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Connected sets.

Fix any number $\delta>0$ and put $A = \{x \in \mathbb{R}: \left|x-3\right|<\delta\}$ and $B = \{x \in \mathbb{R}: \left|x-3\right|>\delta \}$. Prove that $C=A \cup B$ is not a connected ...
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Prove if one set is complete then another set is complete

Let $X$ be a set. Let $l^{\infty}(X,N)$ be all bounded functions on the form $f: X\longrightarrow N$. Let $d(f,g)=\sup\{n(f(x),g(x): x\in X)\}$ be a metric on $l^{\infty}(X,N)$, where $n$ is metric on ...
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Show that $C([0,1],\mathbb{R})$ with the $L_2$ inner product norm is not a Hilbert space.

I need to prove that all continuous functions on the closed set $[0,1]$ is not a Hilbert space. Given the $L_2$ norm. I guess I need to show that every Cauchy sequence in the space, does not ...
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521 views

Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$

Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon >0$ $\epsilon$-chainable Definition (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ ...
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$A = \{(x,y) \in \mathbb{R^2}: y = \frac{1}{x}, x > 0\}$. Show $A$ is closed in $\mathbb{R^2}$

Given $A = \{(x,y) \in \mathbb{R^2}: y = \frac{1}{x}, x > 0\}$. Show (by considering convergent sequences or otherwise) $A$ is closed in $\mathbb{R^2}$. Anyone able to give me some advice on how ...
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Showing $f:\mathbb{R^2} \to \mathbb{R}$, $f(x, y) = x$ is continuous

Let $(x_n)$ be a sequence in $\mathbb{R^2}$ and $c \in \mathbb{R^2}$. To show $f$ is continuous we want to show if $(x_n) \to c$, $f(x) \to f(c)$. As $(x_n) \to c$ we can take $B_\epsilon(c)$, ...
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Showing equivalent metrics have the same convergent sequences

I am trying to show two equivalent metrics $p$ and $d$ on a set $X$ have the same convergent sequences. $p$ and $d$ are such that $kd(x,y) \leq p(x,y) \leq td(x,y)$ for every $x, y \in X$, $k$ and $t$ ...
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89 views

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
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658 views

shows that $d$ is continuous where $d$ is a metric defined on $X$

If $(X; d)$ is a metric space, then the metric $d$ on $X$ induces a product metric $p$ on $X\times X$ by $p((x_1; y_1); (x_2; y_2)) := d(x_1; x_2) + d(y_1; y_2)$ Show that $d :(X \times X ) ...
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Coinciding with the Product Topology

I am a bit confused by this whole question I have in front of me. It defines a distance, $d$, on a product topology $X= \Pi_i X_i $, where $\Pi_iU_i$ forms a basis of open sets and $U_i=X_i$ except ...
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329 views

Meaning of closure of a set

Does closure of a set mean, only adding boundary values if the set is open and leave it as it is if the set is closed?
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139 views

Volume and diagonal length of the Hilbert cube

Here's something sort of fun that I gave thought to a while ago, and now that I've done some maturing mathematically I'm curious to see if my musings are legitimate. Let $H=[0,1] \times ...
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Cantor set - a question about being metrizable and about the connected components

I have a question regarding Cantor set given to me as a homework question (well, part of it): a. Prove that the only connected components of Cantor set are the singletons $\{x\}$ where $x\in C$ ...
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194 views

$C$-doubling $\Bbb R^2$ measure gives measure zero to a straight line?

A metric space $X$ with metric $d$ is said to be doubling on $\Bbb R^2$ if there is some constant $C > 0$ such that for any $x \in X$ and $r > 0$, the Euclidean ball $B(x, r) = \{y:|x − y| < ...
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94 views

Is this the category of metric spaces and continuous functions?

Suppose the object of the category are metric spaces and for $\left(A,d_A\right)$ and $\left(B,d_B\right)$ metric spaces over sets A and B, a morphisms of two metric space is given by a function ...
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Coinciding open sets

I'm given two distances that are defined on some metric space and I need to show that open sets and Cauchy sequences coincide for the two distances. What does this mean? I'm avoiding giving details on ...
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Hausdorff dimension

Could you please give me some hints on (Exercise 1.7.21) of "A Course in Metric Geometry" by Burago, Burago, Ivanov. We have a compact space $X$, which can be written as $X=\bigcup_{i=1}^n X_i$ ...
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117 views

Is {$\phi$} set forms a metric space or not?

Is $\phi$ set forms a metric space or not ? I think, it does not form a metric space, because, we can't specify a metric on $\phi$. But, In many text book, it is not mention that, the set on which, ...
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56 views

At most countable subsets of a compact metric space.

As written, the question is: Let (X,d) be a compact metric space. Prove that for each $\epsilon>0$ there exists a positive integer $N$ such that for each $S \subseteq X$, if $S\thicksim Z_N$, then ...
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224 views

Relationship between sequences and closed sets

I seem to recall that you can say a set is closed if there exists a sequence that converges to a limit point of that set...obviously that is not correct but the idea is that you can deduce a set is ...
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Prove that a compact metric space can be covered by open balls that don't overlap too much.

The problem is: For compact metric space $(X,d)$ prove that for every $r>0$ there exists a subset $S$ of $X$ such that $\{\mbox{Open balls of radius }r\mbox{ centered at }p \mid\mbox{ for all }p ...
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622 views

Closed subset of complete metric space…don't understand last part of theorem.

A closed subset of a complete metric space is a complete subspace. Proof. Let $S$ be a closed subspace of a complete metric space X. Let $(x_n)$ be a Cauchy sequence in $S$. Then $(x_n)$ is a ...
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67 views

Show that $[x, y]$ is complete where $x < y$ in $\mathbb{R}$

As the interval is closed every sequence in the interval converges to some point $x$ in the interval, and every convergent sequence is a Cauchy sequence, hence $[x, y]$ is complete. Is that correct? ...
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100 views

Doubts about metrization theorems

I studying metrization and in different parts I encountered different formulations of theorems, for example in the Nagata–Smirnov metrization theorem I found: A topological space $X$ is metrizable ...
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Showing that a metric space is complete

The Wikipedia page on complete metric spaces gives various examples of metric spaces that are and are not complete - http://en.wikipedia.org/wiki/Complete_metric_space Here's a few lines in ...
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432 views

Equivalent metrics using open balls

Let $d$ and $p$ be two metrics on a set $X$ and let $m$ and $n$ be positive constants such that $md(x,y) \leq p(x,y) \leq nd(x,y)$ for every $x,y \in X$. Show that every open ball for one metric ...
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716 views

Interchanging limit with infimum/supremum

I'm sure I'm having a notational misunderstanding. Anyway, suppose $(f_n)$ is a sequence of continuous functions from a metric space $X$ to $\mathbb{R}$. So, if $(f_n)$ converges uniformly to a ...
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quotient metric spaces for dummies

I was hoping that somebody can explain to me the definition of quotient metric spaces I got the following definition from wikipedia: If $M$ is a metric space with metric $d$, and $\sim$ is an ...
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Existence of minimal subcover for any open cover of a metric space

Suppose (X,d) is a metric space. Does every open cover of X have a minimal subcover with respect to inclusion?
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On $L^p$ and $\ell^p$

If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
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Find the most convenient meeting room

In the following question, one has to find the most convenient x-y-z co-ordinates in a building for a group of employees to sit together. I have tried finding individual average values for x, y & ...
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Help with Metric spaces

$\newcommand{\Int}{\operatorname{Int}}\newcommand{\Bdy}{\operatorname{Bdy}}$ If $A$ and $B$ are sets in a metric space, show that: (note that $\Int$ stands for interior of the set) $\Int (A) \cup ...
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Algorithms for computing or numerically approximating the Prokhorov metric?

I am interested in the following practical question: Given two measures (say those of two parametric distributions), is there an algorithm for computing the Prokhorov metric between them? The general ...
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Find Metric in $\mathbb{R}^2$ s.t. it is not Complete

My friend ask me: How to define a metric in $\mathbb{R}^2$ in such an way that $\mathbb{R}^2$ is not complete. I gave him the following metric: Let $B=\{x\in\mathbb{R}^2:\ \|x\|<1\}$. By a ...
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Why is $L^3$ weaker than $L^2$?

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker ...
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120 views

Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
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Space with non-convergent Cauchy sequence

Not all sequences that are Cauchy are convergent. Here is what I think the example should be. Somehow the metric space is open but does not contain its limit points. Is this the right direction of ...
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281 views

Triangle inequality for hyperbolic distance

A quick way to define the hyperbolic metric in the Poincare disc is via the cross ratio: Given points a,b in the disc, let p,q be the endpoints of the hyperbolic line (halfcircle/line perpendicular to ...
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Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
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123 views

which of he following is a metric space?

$a)$ $C^1[0,1]$ of continuously differentiable real valued functions on $[0,1]$ with the metric $$d(f,g)=\max_{t\in[0,1]}|f-g|$$ I am sure that it is not complete, but could any one help me to ...
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Is $x^n$ Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?

Consider the sequence of functions \begin{equation} f_n(x) = x^n, \quad x \in [0, 1]. \end{equation} Is this sequence Cauchy in $(C[0, 1], ||\cdot||_{\infty})$? The pointwise limit is not ...
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A property of the triangle inequality of Metric Spaces

Question: Assume you have a metric space $(E,d)$ and $A\subset E$ that is nonempty. Define $d_A:x \in E \rightarrow d(x,A)$. Show that $d_A$ is lipschitz and compute its Lipschitz seminorm. My ...
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Open subsets of a complete metric space.

I've been going over previous exams, and I came across a question that I missed. It is as follows: Let $X$ be a complete metric space. Show that every open subset of $X$ is homeomorphic to a ...
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A property about open subsets of compact metric spaces.

I've been looking at various problems from past Topology exams, and I came across a problem dealing with compact metric spaces that I have never seen before. The statment to the problem is as follows: ...
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Discrete Subgroups of $\mbox{Isom}(X)$ and orbits

Let $X$ be a metric space, and let $G$ be a discrete subgroup of $\mbox{Isom}(X)$ in the compact-open topology. Fix $x \in X$. If $X$ is a proper metric space, it's not hard to show using ...
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Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
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Proof that $\|(a,b)\| \leq \|(c,d)\|$ if $0 \leq a \leq c$ and $0 \leq b \leq d$ [duplicate]

Possible Duplicate: Is norm non-decreasing in each variable? Let $\| \cdot \|$ be any norm on $\mathbb{R}^{2}$. Let $0 \leq a \leq c$ and $0 \leq b \leq d$. Show that $\|(a,b)\| \leq ...
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A closed noncompact subspace of a metric space

Say that we have a metric space $X$ and that $Y= \{y_1, y_2,\ldots\}$ is a countable collection of points in $X$ such that for any two points in $Y$, we have $d(y_n, y_m) \geq1$, i.e. the distance ...
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189 views

Notation: Representer Theorem for Reproducing kernel hilbert spaces

Am studying the basic concepts of RKHS and the representer theorem: In $f(x_i)=<f,k(x_i,\mathbb{.})>$, what does $ f$ on the r.h.s denote? What is its structure-is it a vector? I was thinking ...
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Separated sets on a metric space

Let $(X,\rho)$ be a metric space. Two sets $A,B\subseteq X$ are separated if $\overline{A}\cap B=\varnothing$ and $\overline{B}\cap A=\varnothing$. Show that $A$ and $B$ are separated if and only if ...