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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Prove $\{(x,y) \in \mathbb R^2 | 0 < x^2 + y^2 < 1 \}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1 \}$ are homeomorphic to each other

I have $\{(x,y) \in \mathbb R^2 | 0 < x^2 + y^2 < 1 \}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1 \}$ and need to prove they are homeomorphic to each other. I wanted to use the function ...
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81 views

Prove that $\{(x,y) \in \mathbb R^2 | y = x^2 \}$ is not compact

I know I need to choose an open cover and then show it has no finite subcover. If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?
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45 views

Are all continuous bijective translations isometries?

Let $(M, d)$ be a metric space. I define a translation on $M$ to be a function $f$ from $M$ to $M$ such that $d(x,f(x))=d(y,f(y))$ for all $x$ and $y$ in $M$. In a previous question, I asked if every ...
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46 views

Prove the union of compact sets is itself compact (preferably by showing for sequential compactness and then with open covers)) [closed]

Let A$_1$. . . A$_n$ be compact sets in a metric space (X,d). Prove that A$_1$ $\bigcup$ . . . $\bigcup$ A$_n$ is compact
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34 views

Are all metric translations isometries

Let $(M, d)$ be a metric space. I define a translation of $M$ to be a function $f$ from $M$ to $M$ such that $d(x, f(x)) = d(y, f(y))$ for all $x$ and $y$ in $M$. My conjecture is that every ...
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1answer
55 views

Infinite dimensional euclidian space with the product topology metrizable?

Let $\mathbb{R}^{\omega}$ be the space of real sequenes with the product topology. Is $\mathbb{R}^{\omega}$ metrizable?
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42 views

Limit of sums is sum of limits in a metric space

So I'm aware that in a normed space, the limit of the sums is the sum of the limits: For normed space $(X, ||.||)$, if $x_n \rightarrow a$ and $y_n \rightarrow b$, then $(x_n + y_n) \rightarrow ...
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0answers
165 views

Proving existence of uniformly convergent subsequence of a sequence of functions

Let $\{f_n\}_{n \in \mathbb N}$ a sequence of integrable and uniformly bounded functions $f_n:[a,b] \to \mathbb R$ and for each $n$ let $F_n:[a,b] \to \mathbb R$ such that $F_n(x)=\int_a^x f_n(t)dt$ ...
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52 views

Proving that $A_f(x) = \lambda \int_a^x f(t) dt$ is a contraction

Let $C[a,b]$ be the set of all continuous functions $f:[a,b] \rightarrow [a,b]$ where $a,b \in \mathbb{R}$ and $d(f,g)= \max_{x \in [a,b]} \vert f(x)-g(x) \vert $ $\forall$ $f,g \in C[a,b]$. I had to ...
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71 views

Proving that a function is a contraction

The question is: Find values of $a$ such that the function $f(x)=ax^2 -1$ is a contraction on the interval $[1,2]$. I looked up the definition of a function being a contraction on the interval and ...
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1answer
267 views

Prove homeomorphism between $\{(x,y) \in \mathbb R^2 | x^2 + y^2 < 1\}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1\}$

I have two sets $\{(x,y) \in \mathbb R^2 | x^2 + y^2 < 1\}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1\}$ and need to prove they are homeomorphic. I believe I can use the function $f(x,y): ...
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102 views

Closure of equicontinuous family of bounded functions.

Let $B(x,y)$ be the set of all the bounded functions $f: X \to Y$ ($X,Y$ metric spaces). Prove that if $\mathcal F \subset B(x,y)$ is an equicontinuous family, then $\overline {\mathcal F}$ is ...
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1answer
72 views

How to show the set F of all finite sequences is connected in the space c0?

Question: How one can show that the set F of all finite sequences (i.e after n, the entries are zero) is connected in the space c0 (i.e. the space of all sequences that converge to zero) the metric ...
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1answer
65 views

How to show that the space $\mathcal{l}^1$ is connected

How can one show that the space $\mathcal{l}^1$ is connected? $\mathcal{l}^1$ is the metric space of real sequences such that the sum of the absolute values of the entries converges. with metric ...
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39 views

If a set $A$ is disconnected in $(X,d_1)$, then it is disconnected in $(X,d_2)$ for any metric $d_2\geq d_1$

If a set $A$ is disconnected in metric space $(X,d_1)$, then it is disconnected in $(X,d_2)$ for any metric $d_2\geq d_1$ we need to prove or disprove. we think it is true. any open set for $d_1$ is ...
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1answer
53 views

how to show a countable space is totally disconnected for any metric?

Suppose X is countable. We need to show that for any metric d on X the space (X,d) is totally disconnected. It is true that any subset of a countable set is countable. so, divide the space until its ...
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84 views

What's the real name for these things? Categories whose morphisms have “length.”

A fairly obvious "categorification" of metric spaces is as follows. First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose ...
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49 views

Compactness and Convergence of Subsequences

Let $(X,\rho)$ be a metric space. Suppose that $(x_n)_{n\in\mathbb Z_+}$ is such a sequence in $X$ that any subsequence has a further subsequence that is convergent. However, the limits of these ...
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1answer
26 views

Prove that $Lip [a,b]^{\circ}=\emptyset$

Let $Lip[a,b]=\{f \in C[a,b] : \exists k>0, |f(x)-f(y)|\leq k|x-y|\}$, Prove that $Lip[a,b]^{\circ}=\emptyset$ in $C[a,b]$. Suppose there exists $f \in Lip[a,b]^{\circ}$, then $\exists ...
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1answer
59 views

Every finite metric space is discrete

I know that the question is so easy, but I don't know how to conclude. I have that: Let $X$ be a finite metric space. Suppose that $X$ isn't discrete, then exists $x\in X$ such that, $\forall ...
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64 views

Diameters of and distances between sets in metric spaces

I know that: If $\DeclareMathOperator{\diam}{diam}(X,d)$ is a metric space and $A\subset X$ is bounded, then there $\sup \{ d(a,a'):a,a'\in A \}$, called the diameter of the set $A$ and is denoted by ...
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115 views

Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
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21 views

Derivation of Symmetry Property of Metric Spaces

I am given the following modified triangle inequality property of metric spaces, where for any $x_1$, $x_2$, $x_3 \in X$, we have $d(x_1, x_2) \le d(x_1, x_3)+d(x_2, x_3)$. I am tasked to show that ...
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56 views

Show that $\Bbb Q$ and $\Bbb R \backslash \Bbb Q$ with restriction for euclid metrics is not is not complete [closed]

Please someone help me because I can not solve the following example. Show that $\Bbb Q$ and $\Bbb R \backslash \Bbb Q$ with restriction for euclid metrics is not is not complete. Plase help me. ...
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102 views

Find Weight for minimum Manhattan Distance

Let's say, I have three points $(1, 4)$, $(4, 3)$ and $(5, 2)$. I need to find weight $w_1$ and $w_2$ so that the point $(1, 4)$ be the centroid of the points in ...
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58 views

Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
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77 views

Behavior of Hausdorff dimension under homeomorphisms

Let $X$ and $Y$ be metric spaces, $f : X\rightarrow Y$ a homeomorphism. Denote by $\dim_{\mathcal H}$ the Hausdorff dimension. I know that it is possible that $\dim_{\mathcal H} Y < \dim_{\mathcal ...
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47 views

Closed and bounded subset of a metric space which is not complete

I am trying to find a counterexample for a metric space which is not complete and has a closed and bounded subset. Any hint will be helpful. Thank you
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25 views

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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174 views

Examples of rare, meager and nonmeager sets in $\mathbb{R}$

Kreyszig Functional Analysis book presents the following definition. I'm trying to get some examples. (a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so ...
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Tell that sequence $(x_n)$ converges if and only if there $n_0\in \Bbb N$ such that $x_n=x_{n_0}$ for all $n\geq n_0.$*

I do not know how to solve the following example so if any of you can help me solve. Please. The example is as follows: Let $(X,d)$ a discrete metric space and $(x_n)$ is a sequence in $X$. Tell that ...
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1answer
145 views

When is a Lipschitz homeomorphism of metric spaces bi-Lipschitz?

Let $(X,d_X)$ and $(Y, d_Y)$ be metric spaces, and let $f: X \to Y$ be a Lipschitz map which is a homeomorphism of the underlying topological spaces. Are there conditions which assure that $f$ is ...
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38 views

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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1answer
58 views

Comparing different topologies

I have to solve the following: For every function $f\in C[0,1]$, $\varepsilon>0$ and every finite set $A$, set $U(f,A,\varepsilon)$ is defined by $U(f,A,\varepsilon)=\{g\in C[0,1]:\forall x\in A\; ...
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1answer
58 views

strong equivalent metrics

Let $(X,d_x),(Y,d_Y)$ be bounded metric space. Let $f:X\rightarrow Y$ be a homeomorphism. Is it true that there exist $a,b>0$ such that $$ad_X(x_1,x_2)<d_Y(f(x_1),f(x_2))<bd_X(x_1,x_2)$$ for ...
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1answer
60 views

Distance Metric in 4 dimensions $\Bbb R^3\times SO(2)$

The euclidean distance metric, $\sqrt{dx^2+dy^2+dz^2}$, shows the shortest distance between two points in $\Bbb R^3$. What would be the distance metric to show the shortest distance between two ...
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1answer
61 views

find nested closed balls of polynomials s.t. intersection is empty (in a metric space)

the space is the set of all polynomials on $[0,1]$ with metric sup. the space is not complete. we need to find explicitly a nested family of closed balls with radius (of each closed ball) goes to ...
2
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1answer
78 views

Different “$\pi$s” [duplicate]

Does any one know of a concept analogous to $\pi$ in metric spaces. Namely, taking the all the points $1$ away from a point, and measuring the distance as some sort of limit? This was prompted when I ...
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1answer
107 views

Diameter of a subset of a metric space

Let $(\Bbb R,d)$ be the metric space with the metric function $$d(x,y)=\frac{|x-y|}{1 + |x-y|}\;. $$ Calculate $\operatorname{diam}(0,\infty)$. I am thinking the answer is $1$ because ...
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0answers
61 views

The concept of compactness

What is the meaning of, " a metric space is compact" ? I know the formal definition but i don't really understand it's true meaning, what is the concept behind compactness ? I seems like a lot of ...
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2answers
103 views

Two equivalent definitions of convergent sequences?

I know that: Definiton 1. The sequence $(x_n)$ in the metric space $(X,d)$ is said to converge to the point $x_0\in X$ if $$\forall\epsilon>0, \exists n_0\in\mathbb{N} \text{ such that } \forall ...
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1answer
59 views

Cauchy Sequence $\{(x_n )\}_{n=1}^\infty =n$ [duplicate]

Verify that the sequence $\{(x_n )\}_{n=1}^\infty =(n)$ is Cauchy in $R^*$ the metric with $d(x,y)=|\frac{1}{x} -\frac{1}{y}|.$ Please I need help with orgente formal demonstration of this issue.
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86 views

Getting a root of a continuously differentiable function by Banach's Fixed Point Theorem.

Banach Fixed Point Theorem: Consider a metric space $X = (X, d)$, where $X\neq \varnothing$. Suppose that $X$ is complete and let $T: X \to X$ be a contraction on $X$. Then $T$ has precisely one ...
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71 views

How to prove that $(\mathbb{Z}, d)$ with $d(m,n)=\vert m -n \vert$ is complete

I know that the metric space $(X,d)$ is called complete if each Cauchy sequence is convergent, but I don't now how to prove the following: $(\mathbb{Z}, d)$ with $d(m,n)=\vert m -n \vert$ is ...
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2answers
60 views

If $\{x\}$ is an open set in $X$, for all $x\in X$, then all subsets of $X$ is open in $X$

Today for example the teacher ask us to nejdeme next example, but none of us knew, so we left for example homework, but try again but I can not solve the example, so please someone help me, the ...
2
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1answer
165 views

a function is continuous iff the graph is pathwise connected

Let f : [a, b] → R. By the graph of f we mean the set F = {(x, f(x)) : a ≤ x ≤ b} ⊆ R Prove that f is continuous if and only if the graph of f is a pathwise connected subset of the plane. I ...
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2answers
93 views

Metric space question

Let (M,p) be a metric space and suppose that ${x_n}$ is a sequence in (M,p) so that $x_n -> x$ and $x_n->y$. prove x=y Let $E>0$. then, $p(x_n,x)->0$ $lim$ $n->inf$ ...
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1answer
67 views

Proving various properties of metric spaces

Suppose that $p_1$ and $p_2$ are metrics on $M$. Prove that the following are also metrics: (a) $p = p_1 + p_2$ define $p_1(x,y) = |x-y|$ define $p_2(a,b) = |a-b|. p = p_1+p_2 = |x-y|+|a-b|$. But I ...
3
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1answer
135 views

What is the name of this definition of the distance between two probability distributions?

There are many different definitions of the distance between two probability distributions: http://en.wikipedia.org/wiki/Statistical_distance . When P and Q are very similar, many of these converge ...
4
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55 views

Metrics on the plane

Define metrics $\rho$ and $d$ on the plane $\mathbb{R}^2$ as follows: for $x = (x_1, x_2)$ and $y = (y_1, y_2)$, $$\rho(x, y) = |x_1 − y_1| + |x_2 − y_2|\\ d(x, y) = max\{|x_1 − y_1|, |x_2 − y_2|\}$$ ...