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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Proper Proof for Completeness of $\mathbb{R}$ with the Euclidean Metric

My code can't be uploaded because it doesn't work with the websites coding, but here is a pdf of my LaTex code. My question is, is this a proper proof? It feels as if I'm missing something important. ...
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metric on the set of complex sequences

Let X be the set of complex sequences $(a_n)_{n\in\mathbb{N}}\in \mathbb{C}$. Show that the transformation: $$ d((a_n), (b_n)) := \sum_{n=0}^\infty \frac{1}{2^{n+1}} \frac{|a_n - b_n|}{1 + |a_n - ...
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equations for an open ball in a normed space

Let $(V, \|\cdot\|)$ be a normed space. Show that for an open ball $B_1(0) \subseteq V$, it holds true that: $∂B_1(0) = \{x \in V: d(x, 0) = 1\}$ where $d(x, 0) = \|x\|$. Also, figure out the ...
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Is this an existing matrix distance/metric?

I was thinking about comparing different basis transformations and came up with this distance function: $$d(A,B)= \dfrac{||A - B||^2}{||A|| + ||B||}$$ I am using the Schatten-1-norm as the norm here ...
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Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
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the examples of subspace embedding which are not Oblivious

For the definitions of Oblivious Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf. Then, can any one show the examples of subspace embedding which are ...
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29 views

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$? [duplicate]

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$ for the metric induced from the Euclidean metric on $\mathbb{R}$? I think that yes, it is nowehere dense because ...
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the difference and similarity between 'subspace embedding' and ' dimension reduction'

Can someone show me the difference and similarity between 'subspace embedding' and 'dimension reduction' using the mathematical definition? Thanks a lot.
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45 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
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On a question about finite metric spaces

Let $(X,d)$ be a metric space such that every continuous function $f:X\to \mathbb R$ has a finite Image. prove that, $X$ is finite. I tried this: Let $x_0$ be arbitrary element of $X$ and define: ...
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Infinite Intersection of Open Sets [closed]

Give an example of an infinite intersection of open sets which is not open.
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Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
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42 views

Find two disjoint open sets $U, V$ such that $A\subseteq U, B\subseteq V$ where $A,B$ are closed.

Let $A, B$ be two disjoint closed subsets of a certain metric space $(M,d)$. Show that there exist disjoint open subsets $U, V \subseteq M$ such that $A\subseteq U, B\subseteq V$. Give ...
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Metric space analog of the definition of continuity

The definition of continuity in topological spaces is given as: The function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at the point a in $\mathbb{R}^n $ iff given any open ball ...
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constructing an example of a ball of larger radius, contained in a ball of smaller radius

I've found an example where there is a ball of larger radius contained in a ball of smaller radius, but I'm not sure how it works: Let $X = \{ x \in \mathbb{R}^2 : x_1^2 + x_2^2 \leq 9 \}$ with the ...
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$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?

We call two metrices equivalent if for all sequences $x_n,y_n\in\mathbb R$ it holds $\lim_{n\to\infty}d_1(x_n,y_n)=0 \iff\lim_{n\to\infty}d_2(x_n,y_n)=0$ . I have given $d_1(x,y)=|x-y|$ and ...
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Show that any subset of $(\mathbb{N},d)$ is open and closed

Show that any subset of $(\mathbb{N},d)$ is open and closed, where $$d(m,n) = \frac{|m-n|}{1+|m-n|}$$ my attempt: let $A \subset \mathbb{N}$ then for any $x \in A$ we have that $B(x,1/3) = \{x\} ...
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If $E \subset\mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E) \in E$

Suppose that $E \subset \mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E)\in E$. attempt: Suppose $E$ is compact, then $E$ is closed and bounded. Thus $\sup(E)$ and $\inf (E)$ exist. ...
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Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
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Set that is bounded but not totally bounded: Reading textbook

I've been reading a Real Analysis textbook that my friend loaned to me. I have come across a proposition that says that a totally bounded set is bounded, but a bounded set is not always totally ...
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Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...
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Criterion for Isometry

Let $X$ be a topological vector space, with $d$ an invariant metric compatible with the metric. Let $f:X\to X$ be an involutive linear isomorphism. How do you show that $f$ is an isometry? I ...
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Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
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On convergent sequences

Suppose that i have and open and surjective map between two metric spaces $\pi\colon X\to Y,$ and a sequence $(x_n)_{n\in \mathbb{N}}$ such that its image by $\pi$ converges. Is it true that ...
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Intersection of balls in Hamming space [duplicate]

Let $B(x_1, r)$ and $B(x_2,r)$ be balls in $\{0,1\}^n$ (in Hamming distance). Denote by $d$ Hamming distance between $x_1$ and $x_2$. What is $|B(x_1, r) \cap B(x_2, r)|$ (asymptotically)? Upd: I ...
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Example of metric continuous with respect to another metric but generating different topology

Take, say, the standard 2-sphere $S^2$. Equip it with some metric $d$; this metric will generate a topology that may or may not coincide with the standard Euclidean topology. In the case it does, ...
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$X$ is A-space iff the frontier of any closed set in $X$ is compact.

Hi everyone I have troubles with the following proposition: Definition: We say a metric space $(X,d)$ is an A-space iff every Hausdorff image of $X$ under a closed continuous map is metrizable. ...
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Definition of open ball in discrete metric space

I would like some help clarifying the definition of open balls in the discrete metric space. The definition I am provided is: Open balls in the discrete metric space $M = (X,d_0) $ are given by ...
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Prove compact metric spaces $X$ and $Y$ are isomorphic given these conditions [duplicate]

Let $X$ and $Y$ be compact metric spaces and for each finite subset $A$ of $X$ there is a finite subset $B$ of $Y$ such that A is isometric to B and for each finite subset $A$ of $Y$ there is a finite ...
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Prove $x $ is not an element of $E^0$ if and only if $B_r(x) \cap E^c \neq \emptyset $ for all $r> 0$.

Prove: $x \notin E^0$ if and only if $B_r(x) \cap E^c \neq \emptyset $ for all $r> 0$. Proof: I just need help with converse part. Converse: Suppose $B_r(x) \cap E^c \neq \emptyset $ for all ...
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Prove that there exists $y_0\in C$ such that $d(x,y)=\inf_{y\in C} d(x,y)$, i.e. $y_0$ is a closest point to $x$ in $C$.

If $C$ is a closed subset of $R^n$ and $x\in R^n$, prove that there exists $y_0\in C$ such that $d(x,y)=\inf_{y\in C} d(x,y)$, i.e. $y_0$ is a closest point to $x$ in $C$. Here's what I got but ...
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discrete metric, both open and closed.

I've checked several answers though, still don't understand last bit. Taking radius r = 1/2 then every subset is singleton and it is open. But then how do you deduce it is also closed? Well, a ...
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metric spaces byE.T.copson solutions for exersice of chapter 4

M is the set of all analytic function of the complex variable zregular on the unit disc lzl<1 such that sup ( int 0<=r<1
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Show that there exists sets $A, B$ in $R$ such that $(A \cup B)^o \neq A^0 \cup B^o$

$\newcommand{\closure}{\operatorname{closure}}$ Show that there exists sets $A, B$ in $R$ such that 1) $(A \cup B)^\circ \neq A^\circ \cup B^\circ$ and $2)$ $\operatorname{closure}(A \cap B) \neq ...
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“$\sigma$-uniform continuity”

Let $X$ be an arbitrary metric space and $f:X\to\mathbb R$ a bounded continuous function. Is it possible to choose a countable sequence $(A_n)_{n\in\mathbb N}$ of (preferably open or closed) subsets ...
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Closure of $A= (0,1) \cup (1,2)$ vs. Closure of $A = [0,1] \cup \{2\}$

Closure of $A= (0,1) \cup (1,2)$ vs. Closure of $A = [0,1] \cup \{2\}$ I am trying to figure out the difference of the closure of these two sets. Informally, my definition of closure is the ...
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Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
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Does $\sigma$ -compact imply separable?

Let $D$ be a metric space. If $D$ is $\sigma$-compact, does this imply that $D$ is separable? I thought I had a proof, but I think it is wrong. my proof: Let $K_n$ the compact sets such that $K_n ...
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Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...
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Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
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Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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Show restriction map is a contraction/lipschitz mapping

For $C[a,b]$ (set of all continuous real valued functions), define $d(f,g) = \int^{b}_{a}|f(x)-g(x)|dx$ If $[c,d]$ is a subinterval of $[a,b]$ and the mapping $r:C[a,b] \rightarrow C[c,d]$ ...
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Show that a set is not open

Suppose $U_1$ and $U_2$ are both nonempty subsets of $\mathbb R$ such that $U_1 \cap U_2 =\emptyset $ and $U_1\cup U_2 = \mathbb R.$ Consider points $p \in U_1\ \text{and}\ q \in U_2.$ Without loss ...
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How to prove the triangle inequality for this distance?

I'm studying a proof in 'An Introduction to Metric Spaces and Fixed Point Theory' (M. Khamsi, W. Kirk) that shows the equivalence of injectiveness and hyperconvexity for metric spaces. I stumbled over ...
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46 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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Prove that this infinite sum involving metrics is also a metric

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho_i: X\times X\to \Bbb R^+$ with ...
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Prove that this is a metric space

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho: X\times X\to \Bbb R^+$ be a metric on ...
5
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3answers
78 views

Show that $d(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric on $\mathbb{R}$ (triangle inequality) [duplicate]

Prove that $d(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric on $\mathbb{R}$. Definition. A function $d:E \times E \mapsto [0, \infty)$ is called a metric iff whenever $x,y,z \in E$, $d(x,y) = 0$ if ...
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48 views

Compactification of a straight line

Like in the case of mapping a infinite-plane to a sphere (Riemann Sphere), I can understand, that I can map the infinite line ($-\infty,\infty$) to a circle. Secondly, I can also map a finite line ...
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$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?

My textbook says the following: Let $X=[1,\infty)$ and define $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac1x-\frac1y|$ . Then $d_1$ and $d_2$ are inducing the same topology, $(X,d_1)$ is complete but ...