Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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A metric pace is complete if any disjoint closed sets have positive distance

Let $(X,d)$ be a metric space such that $d(A,B)>0$ for any pair of disjoint closed subsets $A,B\subset X$. Show that $(X,d)$ is complete. Suppose $X$ is not complete. Then there exists a Cauchy ...
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If two subsets of a metric space are equal, they have the same limit points $A=B \implies A'=B'$

I imagine this is true by definition, but I don't know how to prove it: If two subsets of a metric space are equal, they have the same limit points. E.g: $A=B \implies A'=B'$ Notation: $Q'$ is the ...
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Show that $(x_k)$ converges to $x = (x_1,…,x_n) \in \Bbb R^n$ if and only if $x_{ki} \to x_i$ as $k \to \infty$ foe each $i$.

Let $x_k = (x_{k1},...,x_{kn}) \in \Bbb R^n$. Show that $(x_k)$ converges to $x = (x_1,....,x_n) \in \Bbb R^n$ if and only if $x_{ki} \to x_i$ as $k \to \infty$ foe each $i$. I can understand the ...
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Proving that ϱ is a metric?

I have an Analysis exam coming soon, and found this practice problem a bit challenging. Any help on this would be appreciated. A metric space $M$ with metric $d$ can always be re-metrized so the ...
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How to find closure of $A$ in $(\mathbb R^2,d)$?

Given $A=\{(x_1,x_2)\in\mathbb{R}^2 : x_1>0\}$ how do I find $Cl_{(\mathbb{R}^2,d)}(A)$? I don't really understand how to tackle a question like this...
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Two metrics having the same property

Let $(X,d_1)$ be a metric space. Assume there are $x,y \in X$, such that $$d_1(x,z)\leq d_1(y,z) \quad \forall z \in X\setminus \lbrace x,y\rbrace. \quad (*) $$ I am trying to show that if one ...
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set c of convergent sequences in the Normed Linear Space of all bounded real sequences under the sup norm is complete

Show that the set c of convergent sequences in the Normed Linear Space of all bounded real sequences under the sup norm is complete my try: I know that the set of all bounded functions from a set X ...
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Intuition for metric space completion

Is this intuition for the completion of an arbitrary metric space (X,$\rho$) correct? I am trying to understand Royden's argument in his Real Analysis book. Construct $\tilde X$ as union of two ...
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Can we make $(1/n)$ converge to any real $r$ w.r.t. a suitable metric ? and other related issues

For every $h \in \mathbb Z$ , I can construct a metric $d_h$ on $\mathbb R$ such that $\Big(\dfrac 1n \Big)$ converges to $h$ w.r.t. the metric $d_h$ , indeed I consider a function $f: \mathbb R \...
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Let $f: \Bbb R \to \Bbb R$ be such that $f^{-1} (a, \infty)$ and $f^{-1} (- \infty, b)$ are open for any $a,b \in \Bbb R$.

Let $f: \Bbb R \to \Bbb R$ be such that $f^{-1} (a, \infty)$ and $f^{-1} (- \infty, b)$ are open for any $a,b \in \Bbb R$. Show that $f$ is continuous. My Try: We first take an arbitary open subset $(...
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Show that the vector space of continuous functions on $X$ is linearly isomorphic to the vector space of all convergent sequences in $\Bbb R$.

Consider the set $X = \{\frac 1n : n \in \Bbb N \} \cup\{\ 0\}$ with the induced metric of $\Bbb R$. Show that the vector space of continuous functions on $X$ is linearly isomorphic to the vector ...
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Prove if $y \in B(x,r_1)$ then $y \in B(a,r)$

Question: For $x \in B(a,r)$ find $r_1$ such that $B(x,r_1) \subset B(a,r)$. That is, if $y \in B(x,r_1)$ then $y\in B(a,r)$. I have tried drawing the following sketch: My attempt: ...
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Example 4.21 in Baby Rudin: How is the map $f^{-1}$ not continuous at the point $(1,0) = f(0)$?

Let $f \colon [0,2\pi ) \to \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}$ be defined as $$f(t) = ( \cos t , \sin t) \ \ \mbox{ for all } \ t \in [0, 2\pi).$$ Then the map $f$ is bijective and ...
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Weak convergence of Dirac measures converges to a Dirac measure?

Let $X$ be a metrizable space and $\{x_n\}$ be a sequence in $X$. Suppose the sequence of Dirac measures $\delta_{x_n} \xrightarrow{w} P$ where $P$ is some probability measure. Prove that $P = \...
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$y \notin E'\implies \exists r\gt 0|B(y,r)$ does not contain any elements of $E$

(For metric spaces in Rudin, related to this answer) Let $E'$ denote the set of all limit points of $E$. If $y \notin E'$, then there must be an $r\gt 0$ such that $B(y,r)$ does not contain any ...
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Infinitely many points in bounded $E$ means infinitely many limit points and a finite number of non-limit points

I have been thinking about limit points and I have thought of the following question: If a set is bounded, and has infinitely many points, are there necessarily a finite number of non-limit points....
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Metric spaces - prove the closedness of a subset of a continuous function

One question in my book is to prove the continuity of a subset of $C[0,1]$ (continuous real functions on [0,1]) with metric $d(f,g) = sup_{x\in[0,1]} |f(x)-g(x)|$. However, I am not sure on how to ...
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The definition of open set in metric space and general topological zpace

The definition of open set is different in metric space and topological space, though metric space is a special case of topological space. The definition in metric space seems to convey the idea that ...
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Compact set and continuous function [duplicate]

Let $(E,d), (E',d')$ be two metric space, and $f:E\rightarrow E'$ an injective function such that the image of any compact set from $E$ is compact in $E'$. How can I prove that $f$ is continuous? ...
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Riemannian metric induced by metric

This seems a very basic and useful construction, and yet I cannot find any reference for it. So my questions are, 1) Is the following definition correct? 2) Is there a simpler construction? 3) Do you ...
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The sequence $(\frac 1n )$ of inverses of natural numbers converges to a limit other than $0$

Finding difficulty to construct a metric on $\Bbb R$ in which the sequence $(\frac 1n )$ of inverses of natural numbers that converges to a limit other than $0$.
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$E'$ is closed, where $E'$ is the set of limit points of $E$

$E'$ is the set of all limit points of $E$. Proving $E'$ is closed: $E$ is finite 1) If $E$ is finite, then $E$ has no limit points and hence $E'=\emptyset$ and hence $E'$ is closed. ...
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Show $x_n$ converges to x* if all convergent subsequences converge to x*

Let $x_n$ be a bounded sequence of real numbers. Show that, if all the convergent subsequence converges to the same limit $x^⋆$ , then $x_n$ is convergent and converges to $x^⋆$ . EDIT: Would this be ...
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How can I find a sequence from $l^p\setminus l^1$?

I am trying to find out how to find this sequence Find a sequence $x$ which is in $l^p$ with $p>1$ but $x \not\in l^1$ Thanks
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Prove there exists a subsequence of the real numbers such that it is monotonically increasing or decreasing.

Let $x_n$ be a sequence of real numbers. Prove that there exists a subsequence $x_{n_k}$ such that either $x_{n_{k+1}} \le x_{n_k} $ for all $k$ or $x_{n_{k+1}} \ge x_{n_k} $ for all $k$. Would it ...
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Interior and closure of a set with metric you can't visualize

$X = \left\{ (x,y) \in \mathbb{R^2} : 0 \leq x \leq 1, 0 \leq y \leq 1 \right\}$ $d((x,y), (x',y')) = \left\{ \begin{array}{lr} 1 & , y \neq y'\\ |x - x'| & , y = y' \end{array} \...
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Prove $\left(\operatorname{Lip}\left([0,1]\right),\lvert\lvert\cdot\rvert\rvert\right)$ is a Banach Space

We denote by $\operatorname{Lip}\left([0,1]\right)$ the collection of all Lipschitz functions on $[0,1]$. We know that a function $f:[0,1] \to \mathbb R$ is called Lipschitz if there exists $K>0$ ...
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alternating proof for completion theorem

Instead of using equivalent class as is used in pugh's real mathematical analysis. If I have proved every metric space $S$ has a isometric copy $S_0$ in $C^0(M,R)$. And since $C^0(M,R)$ is complete, ...
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How can you prove this metric space exercise?

Let $d$ be a metric on $X$. Determine all constants $k$ such that: (i) $kd$ (ii) $d+k$ is a metric on $X$.
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Metric (In $\mathbb{R}^2$) $d$ in which $d((3,3),(4,2))>d((3,3),(3,7))$?

I'd like to find a metric in $\mathbb{R}^2$ (Denoted $d$) in which $d((3,3),(4,2))> d((3,3),(3,7))$. Is there such metric? Adding something. I already have a pseudometric which does that (The one ...
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Application of the Baire category theory

Definition: A set $M\subset X$ is called "of first category" if it is countable union of nowhere dense sets. Otherwise its called "of second category". I want to see whether the following sets are ...
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Any two open balls in $\Bbb R^n$ are homeomorphic.

Any two open balls in $\Bbb R^n$ are homeomorphic. I am finding difficulty to construct a continuous bijective function which can map an open ball to an open ball in $\Bbb R^n$. Will it be easier if ...
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A strange criterion for compactness [duplicate]

Is it true that if every continuous real-valued function on a metric space is bounded, then that metric space is compact
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Homeomorphism of Unit Sphere and Unit Cube

Are the unit sphere and the unit cube in the n-dimensional Euclidean space homeomorphic? If so, can anyone give an explicit formula for the homeomorphism?
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Length shortening Riemannian metrics

I am looking for examples of Riemannian metrics such that the curve length under these metrics are always smaller than the length as measured in Euclidean space. It is just a question that popped into ...
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How do I go about figuring out delta-epsilon proofs?

I'm going through Bert Mendelson's Introduction to Topology on my own. In fact, I've tried to go through it several times and I always get stuck somewhere. This time it's on limits. I think I ...
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Proving a metric space $\mathbb N^{\mathbb N}$ with $d(x,y)=1/\min\{j:x_j\neq y_j\}$ is complete

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= \...
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Chordal Metric - Showing it is in fact a metric

If I have $f(z_{1},z_{2}) = \displaystyle\frac{|z_{1} - z_{2}|}{\sqrt{1+ |z_{1}|^2} \cdot \sqrt{1 + |z_{2}|^2}}$, for $z_{1}, z_{2} \in \mathbb{C}$, how would I show that $f(z_{1},z_{2})$ is a metric?...
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Quasiconformal mappings: Metric deffinition

In the lectures notes http://users.jyu.fi/~pkoskela/quasifinal.pdf (Prof. Koskela has made them freely available from his webpage, so I am guessing is OK that I paste the link here) Quasiconformality ...
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Bounded metric spaces

Suppose that $(X, d)$ is a metric space, $A\subseteq X$. Show that $A$ is bounded if and only if there is some constant $\Delta$ such that $d(a,a')\leq \Delta$ for all $a, a' \in A$.
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Neighbourhood and metric spaces

Suppose that in a metric space $X$ we have $B_r (x)=B_s(y)$ for some $x,y \in X$ and some positive real numbers $r, s$. Is $x=y$? Is $r=s$?
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Determining whats the Induced Metric

I have the normed space $({\rm Lip}([0,1]), \|\cdot\|)$, where ${\rm Lip}([0,1])$ is all Lipschitz functions from $[0,1]$ to $\Bbb R$, and $$\|f\|=|f(0)|+\sup_{0\le x,y\le 1}\frac{|f(x)-f(y)|}{|x-y|}...
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The maximum of two metrics obeys the triangle inequality [closed]

I'm stuck at proving that $\max\{d,d^*\}$ obeys triangle inequality where $d$ and $d^*$ are metrics on the set $X$. Progress I want to show that $$\max\{d(x,y)+d(y,z),d^*(x, y) +d^*(y, z) \} \le \...
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Closure of the set of all matrices with no real eigenvalues in $M_2 (\mathbb R)$

What is the closure of the set of all matrices with no real eigenvalues in $M_2 (\mathbb R)$.
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Asking for some help in proof of the properties of a special type of metric space

Let $(X,d)$ be a metric space such that $d(x,z)\le \max\{d(y,z),d(x,y)\} , \forall x,y,z \in X$ , then is every ball in $X$ both closed and open i.e. is every open ball closed and every closed ball ...
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Example of contraction mapping

Please give me some examples of contraction mapping on $(C[0,1]), \lvert \lvert \cdot \rvert \rvert_\infty)$ and $(C[0,1],\lvert \lvert \cdot \rvert \rvert_1) $. Note that : 1. $\lvert \lvert f \...
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Metric in which $\mathbb{R}$ is not connected.

I'd like to know. Is there a metric in $\mathbb{R}$ in which $\mathbb{R}$ is not connected? Any hint about where could I find any info about that?
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Proving '$A$ open in $V\subseteq M$ (metric space) iff $A=C\cap V$ (certain open $C$ in $M$)'

I want to prove the following: Let $(M,d)$ be a metric space. Let $A\subseteq V\subseteq M$. 1) $A$ is open in $V \Leftrightarrow A = C\cap V$ (for a certain open $C$ in $M$) 2) $A$ is ...
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Metric space with non-metric topology?

Given a metric space $X$, we usually define a topology on $X$ said to be induced by the metric, which behaves very nicely. But my question is: what happens if we define a topology on $X$ other than ...
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Does continuity in product space imply equicontinuity?

I was given a problem, which asked to show that if X is a metric space and if F is a continuous function from the product space (of X with itself) to real, then the family of functions : {F(x,y):x ...