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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Type / Notation of a distance in metric spaces

I've been reading a book and I stumbled upon a notation that I don't understand. Let $(X, d)$ be a metric space. If $A$ is a partition of $X$, then we may consider the metric space $(A, d|_{A \times A}...
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Show at least one limit point

Show that if $r\in\mathbb{R}\backslash \mathbb{Q}$, then $\{e^{i2\pi r n}\}_{n\in\mathbb{N}}$ have at least one limit point I've been sitting with this problem for at while now, but can't figure it ...
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How to turn a NON-strict total order into a strict total order with $R^3$ vectors?

I'm currently working with colored images in the RGB color space. It's trivial to find a ordering in grayscale images (each shade of gray can be though as a value and darker shades comes before than ...
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Definition of Cauchy Sequence

I have a question regarding the definition of a Cauchy sequence of a sequence in a metric space. The definition I learned and that is consistent with Wikipedia defines a sequence $(x_n)_{n=1}^\infty$ ...
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Prove there exists an open set containing a closed set disjoint with another closed set

Let $F$ and $G$ be closed sets in a metric space $X$ and $F \cap G = \emptyset$. Show that there exists an open set $U$ such that $F \subseteq U$ and $\bar{U} \cap G = \emptyset$. I tried proving ...
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Norms That Define An Open Set

How can a norm define a set in a vector space. I don't understand for example how 2 different norms can define a same open set. It's not intuitive to me. An open set doesn't need a norm to be open (...
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Problem regarding isometric isomorphisms [duplicate]

I need help regarding the following two exercises: a) Show that $(\mathbb R^2, d_2)$ and $(\mathbb R,d_1)$ $d_2,d_1$ being the respective euclidean metrics, are not isometric isomorphic, i.e. ...
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Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...
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Showing two norms is not equivalent

Define the norms as $||f||_u=sup_{x\in[a,b]}\{|f(x)|\}\ \ \ \ \ \ \ ||f||=||f||_u+||f'||_u$ Show that $||\cdot||$ and $||\cdot||_u$ is not equivalent I've found a sequence for which $||\cdot||_u$ ...
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Metric on the profinite completion of the integers?

The p-adic integers come with a metric and associated topology, both of which can be restricted down to the integers. Does this also apply to the profinite completion of the integers? Do they have ...
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Metric equivalent same topology

Let $d_1=|x-y|$ and $d_2=|φ(x)-φ(y)|$ where $φ(x)=\frac{x}{(1+|x|)}$ I must proof $d_1$ and $d_2$ define the same topology over $R^2$ I want some hint. just some indication or méthodes .
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Does $f_n(x)=\sqrt{x-a+1/n} $ converge in $C^1$?

Define $f_n(x)=\sqrt{x-a+1/n} $ on $[a,b]\in\mathbb{R}$. Correct me if I'm wrong, but I worked out that: 1) Since $x\geq a$, then $x-a$ is positive (or $0$), and the function is defined for every $...
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All small enough subsets of a compact metric spaces are covered by one element of its open cover [duplicate]

$(X,d)$ is a compact metric space, $\{U_i | i \in I\}$ is an open cover of $X$. I need to show that there is a number $\delta > 0$, such that if $A \subset X$ and diameter of $A \le \delta$, then ...
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128 views

$f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$ is uniformly convergent

I need to prove that this sequence of functions: $$f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$$ converges uniformly on $[0,1]$. This is a question regarding this one. There's an answer ...
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36 views

Find all limit points in metric space

Find all limit point in $(\mathbb{R}^3,d_2)$ ($d_2$ denotes the distance by the eucledian norm) of $\{x_n=(x_n^1,x_n^2,x_n^3)\}_{n\in\mathbb{N}}$ defined by ${x_n^1}=0,1,2,0,1,2,0,1,2...$ ${x_n^2}=...
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Show that $U \subset S$ open in S $\iff$ there exists an open set $U'$ in $X$ such that $U=U' \cap S $

Let X be a metric space and let $S\subset X$ I want to show that $U \subset S$ open in S $\iff$ there exists an open set $U'$ in $X$ such that $U=U' \cap S $ Here is a little bit of my reasonning: ...
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39 views

Prove that a metrizable space is countably compact iff it is compact.

Prove that a metrizable space is countably compact iff it is compact. ($\Rightarrow$) I let $\{O_i\}$ be a countable open cover for $(X,T)$ with a finite subcover. Let $\{U_i\}$ be an uncountable ...
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1answer
102 views

Every 1-Lipschitz function in the closed unit ball has a fixed point

I'm currently trying to solve the following exercise: Let B be the closed unit ball in $\mathbb R^n$ together with the euclidean metric. Show that every 1-Lipschitz function $f:B\to B$ has a ...
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Completeness of a metric space

I need help solving the following exercise: Show that a metric space $(X, d_X)$ is complete if, and only if, for every isometric embedding $f:X \to Y$ in another metric space $(Y,d_Y)$, it holds ...
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Geodesic of metric space

The define of geodesic is in below picture which is from Wiki. I don't know why it is generalize of geodesic for Riemannian manifolds. In fact , I can't see it is the shortest when two point are ...
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$f:x\to N$, $N$ complete, $a\in \overline{X}$. $\lim_{x\to a}f(x)$ exists $\iff$ $d((f(x),f(y))<\epsilon$

Suppose $M$ and $N$ metric spaces, where $N$ is complete. Suppose $f:X\to N$, with $X$ being a subspace of $M$ and $a\in \overline{X}$. I need to show that $$\lim_{x\to a}f(x)$$ exists $$\iff$$ $$\...
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Can a metric space with a finite number of holes be homeomorphic to its completion?

Let $(X,d)$ be a non-complete metric space. Let $\tilde X$ be its completion. Assume $\tilde X \setminus X$ is a finite set. (That is, $X$ has a finite number of "holes"). Is it possible for $X,\...
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$N$ complete metric space, if a sequence of continuous functions $f_n:M\to N$ conv. unif. in a subset $X$, then $f_n$ conv. unif. in $\overline {X}$

I need to prove the following: Suppose $M$ a metric space and $N$ a complete metric space. Show that if a sequence of continuous functions $f_n:M\to N$ converges uniformly in a subset $X$, then $f_n$ ...
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32 views

Does the following function define a distance metric?

For real numeric vectors of length $N$, let $a_n \succ b_n$ be one if true and zero if false. The distance between $A$ and $B$ is $$\sum_1^N a_n \succ b_n$$ Note that this is very similar to the ...
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Any uncountable not-compact topological space has uncountable number of compact and noncompact subsets

Question: Prove that every uncountable topological space which is not compact has an uncountable number of subsets which are compact and an uncountable number which are not compact. Since any finite ...
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What does $\psi \in \mathscr{L}^2(\mathbb{R}^2)^7$ mean?

I'm aware that $\mathscr{L}^2(S)$ usually means the $L^p$-norm over some set in the context of metric spaces. So I guess that the first part "$\mathscr{L}^2(\mathbb{R}^2)$" means a set of $L^2$-norms ...
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How to find the Hausdorff distance between the following two sets?

$A = \{(x,y) \in \mathbb{R}^2: 1 \le x,y \le 3 \}$ and $B=\{(x,y) \in \mathbb{R}^2: (x-4)^2+(x-1)^2 \le 1\}$. I've calculated/guessed that $H(A,B)=\sqrt{8}=\max \{2, \sqrt{8}\}$. Is this right?
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How to prove $d_1(x, y)$ and $d_2(x,y)$ are Lipschitz Equivalent in $\mathbb{R}^n$?

As the title, How to prove $d_1(x, y)=\sum_{i=1}^{n}{|x_i-y_i|}$ and $d_2(x,y)=\sqrt{\sum_{i=1}^{n}{(x_i-y_i)^2}}$ are Lipschitz Equivalent in $\mathbb{R}^n$?
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Can a non complete metric space be homeomorphic to its completion?

Suppose we have a non complete metric space. Can it be homeomorphic to its metric completion?
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Does being Cauchy in $M$ imply being Cauchy in a subspace?

I've just seen an example about the sequence $x_n = \frac{1}{n}$ being Cauchy in $(0,1]$ because "Cauchy in $\mathbb{R}$ $\implies$ Cauchy in $(0,1]$". Is this true for a general metric space $M$ and ...
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Cardinality of a closed uncountable set.

I am aware that every closed uncountable subset of reals has cardinality of the continuum. It's easy to use this result to prove that the same is true in $\mathbb R ^n$, for each $n\in\omega$. ...
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Confusion about open and closed sets in metric spaces.

Let $(X,d)$ be a metric space and $(X,T)$ be the induced topology. I understand that $$B_r(a) = \{x: x \in X \land d(a,x) < r\}$$ is open in $(X,d)$ because it is open in $(X,T)$. I'm trying to ...
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Prove that a closed ball $B$ is closed and bounded in $(C[0,1], d^*)$

Let $C[0,1]$ be the set of all continuous functions from $[0,1]$ into $\Bbb R$ and let $(C[0,1], d^*)$ be the metric space defined as $$d^*=sup\{|f(x) - g(x)|: x \in [0,1]\}$$ Let $$B = \{f: f \in C[...
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$\inf_{x\in A}{\limsup_nd(x_n, x)} = \limsup_n[\inf_{x\in A}d(x_n, x)] $ for compact subset $A$.

let $ (X, d) $ be a complete metric space, $ A\subset X $ be compact and take a sequence $ (x_n) \subset X $\ $ A $ as a bounded sequence. Since infimum is independent from n , does the following ...
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If $\ d(x_n,x) $ exist then $\ (x_n) $ must be converge a point in $ X $ ?

Let $\ (X,d) $ be complete metric space, $\ x \in X $ and $\ (x_n) \subset X $ bounded sequence. If the real valued sequence {$\ d(x_n,x) $} convgergent then $\ (x_n) $ must be converge a point in ...
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Bounded in $\Bbb R^n$

Let $n$ be a positive integer, $d$ the Euclidean metric on $\Bbb R^n$, and $X$ any subset of $\Bbb R^n$. Prove that $X$ is bounded in $(\Bbb R^n, d)$ iff there exists an $M > 0$ such that for all $...
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Convergence in metric space and show cauchy sequence in metric space

I consider a metric space ($C^1(\mathbb{R}), \mathbb{R}$), $\Vert \cdot \Vert_u$) where $\Vert \cdot \Vert_u$ is the uniform norm, and I want to show that for $f_n \in C^1$, that is the continous ...
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Slicker way to prove $\rho$ is a metric

Let $d$ be a metric on $X$, and define $\rho: X^{2} \to \mathbb{R}$ as $$\rho(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ The difficulty is in checking the triangle inequality. So, I can prove this by writing $f(t)...
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Every separable metric space has a countable base

A collection $\{V_{\alpha}\}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x\in X$ and every open set $G\subset X$ such that $x\in G$, we have $x \in V_\...
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Show that exist a unique $x \in \bigcap_{n \in \mathbb{N}}{A_{n}}$.

Show that if $A_{n}$ is a sequence of closed, not empty and bounded sets of a complete metric space $M$ and $\operatorname{diam}(A_{n})$ vanishes, then exist a unique $x \in \bigcap_{n \in \mathbb{N}}{...
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Ball in $\mathbb{R}$ and limits

Let $q \in \mathbb{R}^2$, $D \subset \mathbb{R}^2$ such that there exists $r > 0$ with $B(q,r) \setminus \{q\} \subset D$. Let $f$ be a real-valued function defined on $D$ which satisfies: $\lim_{...
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Proving that a sequence is Cauchy 5

We want to prove that the sequence $a_n = n^2$ is Cauchy in the metric space $(E, d)$, with $E = [0, \infty[$ and $d(x, y) = |\arctan(x) - \arctan(y)|$. I proceed in the following way: $a_n$ is ...
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Unique projection onto nonconvex sets [duplicate]

If $C \subseteq \mathbb R^n$ is closed and convex, then for every $x \in \mathbb R^n$, there is a unique $y \in C$ minimizing $d(x,y)$. For what other subsets $C$ does this property hold? Would ...
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Question about the proof of compact subsets of metric spaces are closed and bounded.

If $A$ is a compact subset of a metric space $(X,d)$ then A is closed and bounded. What I'm confused about is this part of the proof: Let $x_0$ be fixed and define the mapping $f:(A,T) \rightarrow \...
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Prove that any discrete metric space is complete [duplicate]

First I know it have been asked several times, but I cannot seem to understand the prove for any discrete metric space is complete. Are there someone who maybe can write a very detailed prove? Thank ...
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$\ell^p$-spaces for $p<1$

It is well known that whenever $p\in (0,1)$, the mapping $$ d_p(x,y):=\|x-y\|_{\ell^p}:=\left(\sum_{n=1}^\infty |x_n-y_n|^p\right)^\frac{1}{p} $$ turns $$\ell^p:=\{(x_n)_{n\in \mathbb N}:\|x\|_{\ell^...
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36 views

prove $||A_n^{-1}||$ is bounded

Let $(A_n)_{n=1}^\infty{} \in GL_{n\times n}(\mathbb{R}) $. $\lim_{n\rightarrow\infty} A_n = A$; $A\neq0$ is invertible. I have a notion that for any norm $$||\cdot||:GL_{n\times n}(\mathbb{R}) \...
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An elementary question about real plane metrics

Given the metric $d_p$ on the real plane, i.e., $ d_p(x,y)=[|x_1-y_1|^p+|x_2-y_2|^p]^{1/p}$ For which values of $p \geq 1$ is it true that the following set is the usual line segment in the real ...
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Hausdorff distance on power sets

Consider a general metric space $(S,d)$, with $d$ a $1$-bounded metric, and let $X,Y \subset S$ be two closed subsets of $S$. Notice that $X$ and $Y$ are not compact. Let $\mathcal{P}(X)$ denote the ...
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34 views

When are the distance between points and sets well-defined?

Let $G$ be an open subset of $\mathbb C$. I would like to prove that this set $\{z\in G; d(z,\mathbb C-G)\ge 1/n\}$, where $n\in \mathbb R$, is well-defined. In another words, I would like to know if ...