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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Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
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Continuous function in metric spaces

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. $f(x): (P,\rho) → (\mathbb R,\rho_1): 0$ if $x∈[-1,1]$ or $1$ if $x∈\mathbb R/[-1,1]$. $\rho_1 (x,y) = \sum_{k=1}^\infty |x_k-y_k|$. ...
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Non-constant Cauchy sequence

I need to find an example of non-constant Cauchy sequence in $\mathbb E^2$. The metric in question is $\rho_2$, so Cauchy sequence would be sequence for which following is true: $\sqrt {(x_m - y_m)^2 ...
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Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
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Prove that the distance function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ $0<p<1$ is a metric on R^n

Hi I am trying to prove that for $0<p<1$ the function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ is a metric on $\mathbb{R}^n$. I am struggling with the triangle inequality part; We have to prove ...
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A continuous function on the real line such that the preimage of every point is either empty of has exactly 3 points [duplicate]

Let $f : \mathbb{R} \to \mathbb{R}$ be a function with all fibres $(\lbrace{x \in \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c)$, $c \in \mathbb{R})$ either empty or consisting of exactly three points. ...
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Proving that $b \in \overline{A}$ if and only if $\rho(b,A) = 0$

I need some help with this problem: Let be $(X,\rho)$ a metric space, $A \subseteq X$ and $b \in X$. The distance from $A$ to $b$ is defined as $\rho(b,A) = \inf\,\{ \rho(b,a) : a \in A \}$. Prove ...
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Finitely many connected components, prove interiors are also connected

Show that in a space with finitely many connected components $C_i, i = 1, ..., n$ their interiors $Int(C_i)$ are also connected. Is it true in general that the interior of a connected component is ...
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Showing a metric space is complete.

On the space of continuous functions on $[0,1]$, I have a metric $$d(f,g) = \sup | \alpha(x) (f(x) -g(x))|,$$ where $\alpha(x)$ is a continuous function and $\alpha(x) \ne 0$. I'm trying to find ...
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Group of distances

How to prove that $$g:\Bbb R^3 \to \Bbb R^3 \in G = \{g \, | \, \text{ for each } g \text{ exists } n\in\mathbb{Z} : r(g(x),g(y)=2^n r(x,y) \}$$ for each $x,y\in \Bbb R^3, r$ is an euclidean ...
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Equivalent of $\ln\ln(N(\epsilon))$ where $N(\varepsilon)$ is the minimum of balls for covering $A$.

Let $E=\mathcal{C}^0([0,1],\mathbb{R})$ with the uniform convergence and $$A=\{f\in E\ |\ f(0) = 0\text{ and } \forall x,y\in[0,1]\ |f(x) - f(y)| \leqslant |x-y|\} $$ For $\varepsilon >0$ ...
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Counterexample $||S+T||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}+||T||_{\mathbb{B}(E)}$

Let $S,T\in \mathbb{B}(E),\ \mathbb{B}(E)=\left\{T:E\to E:T\ linear\ bounded\right\}$ Give a countraexamples such that: (a) $||S+T||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}+||T||_{\mathbb{B}(E)}$ ...
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About interior of the frontier (proof-checking)

Let $M$ be a metric space, and $A \subset M$ an open set. Show that $\stackrel{o}{\widehat{\partial A}} = \emptyset$. ($\stackrel{o}{\widehat{\partial A}}$ is the interior of the frontier) I ...
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Compatible maps

Definition Let $M$ be a subset of a metric space $X$ and $T, I :M\to M$ be $M$-invariant maps. Then the pair $(T,I)$ is called compatible if $$\lim_{n\to\infty}d(ITx_n, TIx_n)=0$$ whenever ${x_n}$ is ...
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Examples of decreasing sequences of closed sets with constant diameter and empty intersection in complete metric spaces

Looking through older exams from the topology class I'm taking, I found an interesting problem. Give an example: $ (X, d) $ - a complete metric space $ F_1 \subset F_2 \subset F_3 \subset ... $ - a ...
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Why must an interior point of $E$ be an element of $E$?

This question takes place in a general metric space $X$. Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. This is like the ...
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Closed Sets and Open Sets

I have a few questions regarding open and closed sets. I am given a set: $$A = \left\{ \frac{1}{x}: x \in \mathbb{Z}^+ \right\},$$ I was asked to find the interior, closure, and boundary points. This ...
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In a metric space, prove there is an invertible function $\Bbb R^n\to\Bbb R^n$ such that $f(a)=b$

I would like to prove the following theorem from Mendelson's Introduction to Topology: For each $a,b\in\Bbb R^n$, prove that there is a topological equivalence between $(\Bbb R^{n},d)$ and ...
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All metrics on finite spaces are equiv - I'm happy with this, except for that annoying point metric.

Okay for $\mathbb{R}^2$ say, I'm quite happy that $d_1(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ and $d_2(x,y)=\max(\{|x_1-y_1|,|x_2-y_2|\})$ that the unit ball is a circle in one and a square in the other ...
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Simple openness/closedness question

I read on another thread here that the set $\{0\}$ is open in $\{0,1\}$, with $\{0, 1\}$ a subset of $\textbf{R}$. This makes sense to me b/c $\exists$ an open set in $\textbf{R}$, say, $(-1,1)$ s.t. ...
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Is $x_n=\frac{1}{n}$ Cauchy sequence with the metric $d(x,y)=|\arctan x−\arctan y|$? [closed]

Let $(\Bbb{R},d)$ be a metric space where $d(x,y)=\left\vert\arctan x−\arctan y\right\vert$. Is the sequence $x_n=\frac{1}{n}$ a Cauchy sequence with this metric? The definition of Cauchy ...
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Is d(0,a) = d(b,a+b) true in general?

As an exercise, I was trying to prove, given a distance metric $d$ on a metric space $X$, that if $a,b\in X$ then $d(0, a) = d(b, a+b)$ but I'm not seeing a way to do it. Is this necessarily true in ...
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416 views

Can a sequence of functions converge to a discontinuous limit under norm?

I'm a bit confused about how to take the distance between two functions where one function is discontinuous. Supposing we have the $L^1$ metric $d_1$ and $f_n(x) = x^n$ defined over $[0, 1]$. $x^n$ ...
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What is meant by gluing two metric spaces together?

"Gluing" constructions are common in topology: by gluing two disks along their boundaries we get a sphere; by gluing a cylindical "handle" to a sphere we get a torus, and so forth. If the original ...
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Metric on a finite set

If $X$ is a finite set and d is an arbitrary metric. Prove that $(X, d)$ is complete. My solution: Let $X =$ {$x_1, x_2, ... , x_n|n \in \mathbb{N}$} $\exists N\in \mathbb{N}$ s.t. $x_n = x$ for ...
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When does a descending sequence of nonempty sets have a non empty intersection?

Let $\langle F_n\rangle_{n\in\Bbb{N}}$ be a descending sequence of nonempty sets in a Metric Space - $F_1\supset F_2\supset\cdots$. What are the conditions on the underlying space so that ...
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Can I define a ball using a distance which is not metric?

I define X as a set with say 50 elements. I manually assign distance between any to elements using numbers from the irrational number pi. So 50*(50-1)/2 distance values are assigned. Now for each ...
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Existence of fixed point

I will copy the definition I am using just to make things clearer. Def. Let $(X,d)$ be a metric space and let $F:A(\subset X)\rightarrow X$. We say F is a contraction if there exists $\lambda$ where ...
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Hyperbolic metric spaces

I'm trying to prove a simple proposition wich is in Burago's "A Course in Metric Spaces" (Exercise $8.4.5$, p.$287$). Before exposing my problem, let me give some definitions. A metric space $(X,d)$ ...
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Limits in cofinite topology/why is the limit of x_n = n equal to 1 in the cofinite topology.

Just reading about topological spaces for my exam, and I was wondering if anybody could explain exactly how limits work in the cofinite topology. So I am aware of the topological definition of a ...
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Metric space and closed sets (book misprint?)

I am not sure if there is a misprint in this corollary or if I am not getting the idea right. Corollary. Let $X$ be a metric space and let $A\subset X$. Then A is closed in $X$ iff: $$ ...
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proving that closed subspace of complete metric space is complete

$4.9$ Let $A$ be a subset of a metric space $S$. If $A$ is complete, prove that $A$ is closed. Prove that converse also holds if $S$ is complete. For the first part, I assumed $\{ a_n\}$ to a ...
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If $d$ is a metric, is $d(x,y)/(1+d(x,0)+d(y,0))$ a metric?

I now that one can show that if $d$ is a metric on a vectorspace $X$ then so is $$\varrho(x,y):=\frac{d(x,y)}{1+d(x,y)}.$$ This easily follows from the fact that the function $s \mapsto \frac{s}{1+s}$ ...
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Distance between any two points in a compact metric space

I am given the following problem: Show that if a metric space (X,d) is compact (meaning X is compact with respect to the metric d), then there exist points a,b ∈ X such that d(a,b) = ...
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Proving that there is no norm for the space of real-valued sequences making it a complete metric space.

Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become ...
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Simple examples of proper metric spaces?

I've encountered the term of a "proper" metric space(a metric space is called proper if every closed, bounded subspace is compact), which struck as quite an interesting one, but I can't find any good ...
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How to prove that a sub-space of the functions $f: X \to Y$ is equicontinuous?

Let $X$ and $Y$ be two metric and compact spaces, and $C(X,Y)$ - the metric space of the continuous functions $f:X\rightarrow Y$. Denote by $Y^X$ the space of all functions (not just continuous) ...
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Conditions for Metricization of Cartesian Product of Metric Spaces

Let $M_1$ and $M_2$ be metric spaces with metrics $\rho_1$ and $\rho_2$ respectively. What are some necessary and sufficient conditions on $f:\mathbb{R}_{+}^2\to\mathbb{R}_{+}$ that make ...
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What makes metric spaces special?

This is not a question about what is special about a metric space in itself; instead, I'm wondering what sets metric spaces apart from uniform spaces? An explanation is in order. As a parallel, when ...
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If two metric spaces are homeomorphic, do their completions have to be homeomorphic?

Let $ (X_1, d_1) $ and $ (X_2, d_2) $ be metric spaces and $ (X_1^*,d_1^*), (X_2^*,d_2^*) $, respectively, their completions. If $ X_1 $ and $ X_2 $ are homeomorphic, then so are $ X_1^* $ and $ ...
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About equivalent norms

Consider $E$ the space of the functions $f: [0,1] \to \mathbb{R}$ such that $f(0) = 0$ and $f$ satisfies a Lipschitz condition. We define two norms: $$\|f\| = \sup_{x \in [0,1]} |f(x)|$$ and ...
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Distance in metric space p_{1}

I need to evaluate distance of point [6,6] and circle $x^2 + y^2 = 25$ in metric space $p_{1}(x,y) = ∑|x_k-y_k|$ (sum metric). I know that I need to count $inf(p_{1}([6,6],X), X $ are points from ...
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Proving a metric with absolute value [duplicate]

I need to prove that function $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is ...
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Write an open set in terms of a closed set

$(X,d)$ is a metric space. We fix a point, $a \in X$, and we let $A = \bigcap_{n\in\mathbb{N}} \left\{x: d(x,a) < r + \frac{1}{n} \right\} \in X$. Is $A$ open or closed? If it is closed. What is ...
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A Fundamental Property of Metric Spaces …

Let $(X,d)$ be a metric space and $A\subset X$ and also suppose that $G$ is open in $X$ prove the identity: $$ \overline {G\cap A}=\overline {G\cap \overline A} $$ Proposition: The intersection of ...
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Open ball of radius, r = 0 is empty?

Is $B(a;0) = \{x : d(a, x) < 0\} = \varnothing$? And if so, is it always the case? The reason I ask is because I want to know if the open interval $(a,a) = \varnothing$ when $a \in \mathbb{R}$. ...
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Coupling methods

Distance between probability measures Let $(X,d)$ be a compact metric space, and let $\mu$ and $\nu$ be two probability measures on $X$. We can define the Wasserstein distance between $\mu$ and $\nu$ ...
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Natural example where $\ell_\infty$ distance appears.

The $\ell_2$ distance has a natural connotation: the straight line distance between two points "as the crow flies". Similarly, the $\ell_1$ distance has a natural connotation: the length of a path ...
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What does it means that sequences characterize closed sets and functions?

A text book I'm reading says at one point the following: "In metric spaces are sequences the ones which chacterize closed sets and continuous functions". What is exactly the meaning of that ...
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What is the completion of this space?

This question asks us to show that $\Bbb R$ with the following metric is not complete: Fix a strictly positive function $f \in L^1(\Bbb R)$, and let $d(x,y)=\left|\int_x^y f(t)dt\right|$. It's easy ...