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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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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Is this 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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63 views

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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58 views

A subspace of a separable metric space is separable [closed]

Prove that a subspace of a separable metric space is separable my attempt to solution is Let $X$ be a separable metric space take $A \subset X$ i need to show that A is separable i.e show that $A$ ...
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set similarity metric

Suppose I've two sets: $S_1 = \{a, b, c\}$ and $S_2 = \{b, c, d\}$. Each element in the set is associated with a real number between 0 and 1 (could be seen as its probability of presence). Example: ...
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54 views

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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65 views

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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53 views

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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25 views

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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374 views

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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72 views

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 ...
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Prove that $\{n\}$ is a Cauchy sequence that doesn't converge.

Consider the distance function given by $d:\mathbb{R}\times\mathbb{R}\to\mathbb{R},\;d(x,y)=|\int_x^yf(t)dt|$ where $f$ is a continuous and positive function such that $\int_{-\infty}^{+\infty}f$ ...
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Closest distance metric to cosine similarity

I have a model of data that produces vectors that are similar in terms of their cosine similarity. Because this is the output of a complex process, I don't have much insight into why the cosine ...
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55 views

Topology: Open, Closed Set and infinity norm

Since last week I've been learning a bit about Topology in Calculus and know the basic definitions of open, closed, norm, etc. Now I try to solve this question but I don't know how to. Its really ...
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Some Property of Cantor set?

Draw a Cantor set $C$ on the circle and consider the set $A$ of all chords between points of C. Prove that $A$ is compact. Is $A$ convex? The proof of first part goes as follows: As we know ...
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29 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
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76 views

Topology. Why is $T^{-1}$ continuous?

Today we did this proof, but we could not finish it and our prof said that the end would be easy, but I could not finish this proof. Let $X$ be a $T_3$ space with a countable basis $B$. Then we ...
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32 views

a decreasing sequence of convex domains

Let $\Omega_1 \supset \Omega_n \supset\cdots$ a decreasing sequence of bounded, convex and open sets in $R^n$. Define $\Omega = \operatorname{int} \left(\overline{\bigcap \Omega_n}\right)$ and supose ...
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Show with an exemple that the inclusion could be real

Im trying to Solve this problem. I have started to Solve the first part, but need to show it with an example. Thankfull for help.
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Show graph is closed if f is continuous

I need help figuring out how to complete this proof: Show that the graph of $f$ is closed if $f$ is continuous.
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Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
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Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
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Compact metric connected space

If I have a compact metric space $X$ such that for all $a,b \in X$, there are points $a:=x_1,...x_n=:b$ such that $d(x_i,x_{i+1})< \varepsilon$, then this space is connected. Somehow, I don't see ...
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Proof metric space with distance function

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii). Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := ...
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Comparing a vector with a sorted vector

I am currently looking at a database in which the tuples have a natural order in which each tuple has an integer ID that reflects its age. The more recent the tuple, the larger its ID. Now we are ...
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Hausdorff distance in Cantor set

I need help on this problem.. Let $C_n$ denotes the nth stage in the construction of the Cantor ternary set, i.e. $C_0=[0,1]$, $C_1=[0,1/3] \cap [2/3,1]$ and so on. Find the Hausdorff distance ...
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How to prove that the French railways metric is a metric

I am trying to work just one case of showing that the French railways metric defined by a metric space $(\mathbb{R}^2,d)$ is actually a metric: $$d(x,y) = \begin{cases} \|x-y\|, & \text{if ...
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The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite ...
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Completeness of the metric r(x,y)=min{d(x,y},1}

Can somebody help me out with the following: If (X,d) is a metric space and r(x,y)=min{d(x,y},1} for all x,y in X. I proved that r is a metric and that r and d are equivalent. Now I want to prove ...
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64 views

Proving a metric induces the product topology

Let $(M,d)$ and $(N,d')$ be metric spaces. Prove that the product topology is induced by the metric $d_1((x,y),(x',y')=d(x,x')+d(y,y')$ and ...
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33 views

Inclusions in Metric Topology

Prove that if $\exists L>0 : d_2(x,y)\geq Ld_1(x,y)$ then the topology induced by $d_2$ is finer than the topology induced by $d_1$. I'll call $T_1$ the topology induced by $d_1$ and $T_2$ the ...
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Can't prove statement jumped over in book proof, I actually think it might be wrong. But it is used throughout the book

I wish to prove that the metric $d(x,y)=\sum^\infty_{n=1}\frac{1}{2^n}\frac{|x_n-y_i|}{1+|x_n-y_n|}$ This is a metric on infinite sequences, x,y and z are sequences with terms $x_i$ and so forth. ...
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prove that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ w.r.t these metrics

Prove that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ w.r.t these metrics: $d_{2}(x,y)=(\sum_{k=1}^{n}(x_{k}-y_{k})^2)^\frac{1}{2}$ $d_{0}(x,y))=\max_{1\leq k\leq n}|x_{k}-y_{k}|$ my attempt of ...
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help me please proof about contiunity of metric

metric is a continuous function.I need to prove that.But I cannot show that if $|x-a|<\delta$ then $|d(x,y)-d(y,a)| < \varepsilon $. for any metric $d$ $|x-a|<\delta$ is it true? or Should ...
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Triangle inequality to prove metric

I'm trying to prove that $$d(p,q)=2\sqrt{\left\vert p-q\right\vert} $$ is a metric. Now the first axioms are quite obvious, but I'm having difficulties trying to prove triangle equality for this. It's ...
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Continous function on compact interval - bounded

Let $K$ be a compact interval in $\mathbb{R}$. Then every continous function $\phi :K\rightarrow \mathbb{R}^d$ is automatically bounded. Is this a consequence of; the image of a compact is compact ? ...
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Possible forms of open balls

Consider $X= ( \Bbb Q \cap [ 0,3] , d_E)$The question is as such: "Describe the possible forms that an open ball can take in $X = (\Bbb Q ∩ [0, 3], d_E )$." I don't understand this means exactly. ...
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Creating a metric from a pseudometric

Given the following definition of a pseudo-metric on the set $X$ : A pseudo-metric on the set $X$ is a map $d:X \times X \to \Bbb R^+$ such that for all $x ,y \text{ and } z \in X :$ (PM1) $x=y ...
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Is it true that $0<0$ is a part of the proof for a unique fixed point of a contraction map?

I think it isn't, but my friend told me that this statement is true, he said that $0<0$ is even part of the proof for a unique fixed point of a contraction map. I google it but I couldn't find ...
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Hausdorff is a metric outer measure

I am new to measure& hausdorff measure, when looking at the proof of this property, I have a question : Given $E_1,E_2 \subset X,X$ is a metric space, we want to prove that if ...