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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What is the relation between convex metric spaces and convex sets?

Here's another question that came to mind when I was reading the article on convex metric spaces in Wikipedia: According to the article, "a circle, with the distance between two points measured along ...
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Metric of space of plane curve

I am looking for a metric $d$ for smooth 2D curves. Hence $d(x,y)$ is the distance between the curves x and y. For the moment, we may assume that $x$ and $y$ are just directed line segments. Do you ...
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The space of continuous, bounded functions from a metric space $X$ to $\mathbb R$

Let $(X,d)$ be a metric space. We denote by $C_b(X;\mathbb{R})$ the space of continuous and bounded functions from $X$ into $\mathbb{R}$, equipped with the sup-norm metric. We define a mapping $O: X ...
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Why is Hausdorff metric defined this way? [duplicate]

From Wikipedia The definition of the Hausdorff distance can be derived by a series of natural extensions of the distance function $d(x, y)$ in the underlying metric space $M$, as follows:[4] ...
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Critique this proof on compactness.

Problem: Prove or disprove, the metric space $X$ containing infinitely many points with the discrete metric is compact. Write a proof in the language of sequences and covers Proof: Take $(1/n) \to ...
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Show that $\rho(x,y) = |\sin(x)-\sin(y)|$ is not a metric on $\mathbb{R}$?

Show that $\rho(x,y) = |\sin(x)-\sin(y)|$ is not a metric on $\mathbb{R}$ and in what condition must be imposed on a function $f:\mathbb{R}\to\mathbb{R}$ in order for $\rho(x,y)=|f(x)-f(y)|$ to be a ...
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Show that there is a compact neighbourhood $B$ of $x$ such that $B \cap F = \emptyset$.

Let $X$ be a compact Hausdorff space, $F \subset X$ closed and $x \notin F$ . Show that there is a compact neighbourhood $B$ of $x$ such that $B \cap F = \emptyset$. I'm trying to use the fact that ...
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Show that for each $a\in [0, \infty)$, the subspace $[a, \infty)$ is also compact

i) Show that the intervals $(a, \infty)$, $a \in (0, \infty)$ together with $\emptyset$ and $[0, \infty)$ form a topology on $[0, \infty)$. ii) Show that in this topology $[0, \infty)$ is compact. ...
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$\operatorname{dist}(A, B) = 0 \land A \cap B = \emptyset \implies \partial A \cap \partial B \neq \emptyset$

A similar question: Distance of two sets and their closest points The question above, however, defines distance differently. The definition we work under is: $$\operatorname{dist}(A, ...
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Understanding a claim — which interpretation is right?

I'm trying to disprove the following claim. If $A$, as a subspace of $X$, has discrete topology, then $X$ has discrete topology. The statement ...
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Metrics with infinite distances.

I've been wondering about the spaces $\Bbb R\cup\{-\infty,+\infty\}$ and $\Bbb C\cup\{\infty\}.$ Is there a useful generalization of the definition of a metric they satisfy? I thought it would be ...
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A metric such that $d(0,1)>1000d(0,2)$

I need to find a metric on $[0,2]$ such that $d(0,1)>1000d(0,2)$ Here is the the example I came up with the following $d(x,y)=\begin{cases} 0\ \ x=y \\ 1 \ \ x\neq y+1\\ 1001 \ \ x=y+1 ...
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Proof is contradicting what I know to be true, what is wrong?

I'm trying to prove that a given function is a metric on some set. I'm confused now because the maths is not not adding up. Let $X=${$x \in \mathbb{R^2}:\lvert x \rvert =1$}. Given $x,y \in X$, ...
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Does continuous convergence imply uniform convergence?

Question Related to a nice problem I met yesterday, a question arises: Suppose $\{f_n\}$ is a sequence of mappings from a connected complete metric space $X$ to a metric space $Y$. Given $f\colon ...
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Why is a rectangle not a neighborhood of its corners?

I'm trying to puzzle out a statement given in the Wikipedia article on topological neighborhoods, which uses this definition: If $X$ is a topological space and $p$ is a point in $X$, a ...
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Finding an isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$

Prove the isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$, where $B$ is the collection of all bi-linear mappings. In order to do so, give a natural isomorphism between ...
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How do I sketch the following metrics:

In $\mathbb{R}^2$ sketch $B((1,2),3)$, the open ball of radius $3$ at the point $(1,2)$, with the following metrics: a.) the post-office metric given by $$d(x,y) = \left\{ \begin{array}{l l} ...
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Does a norm have to map to $\mathbb R$?

Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary. ...
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Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric. [duplicate]

where $(X,d)$ is a metric and $x,y \in X$. I know we need to show: non-negativity: $d(x,y)\geq$ 0 $d(x,y)=0$ if and only if $x=y$ symmetry: $d(x,y)=d(y,x)$ $d(x,z)\leq d(x,y) + d(y,z)$ I think we ...
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How to show that a point is not an interior point?

I understand that in order to show that a point, $x$, is an interior point of some set $A \subset B$, where $(B,d)$ is a metric space you just need to show that you can have an open ball around $x$ ...
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If $f_n(x)=x^n$ converges to $f$, why is $f$ not continuous?

I was reading my Analysis course notes and had some trouble. I hope you can help me. Let $C(X)=\{ f | f:X \longrightarrow \mathbb{R} \text{ is a continuous function}\}$. It was already stated and ...
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Metric Spaces Analysis

Let $(X,d)$ be a metric space and for $x,y \in X$ define $d_b(x,y) =$ $ \dfrac{d(x,y)}{1 + d(x,y)}$ a) show that $d_b$ is a metric on $X$ Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$ ...
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Metric induced Topology

The Problem: Given a metric space $(X,d)$, define a new metric $d'$ on X by $$ d'(x,y)=\frac{d(x,y)}{d(x,y)+1} $$ Is the topology induced by $d'$ the same as the topology induced by d? Prove or ...
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Closure in Metric Space

I need help understanding Theorem 2.27(c) in Rudin. If $X$ is a metric space and $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\bar ...
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Isometric involutions and sections

I have a metric space $X$ and an isometric involution defined on it $i:X\rightarrow X$. My intuiton tells me that I can find a (continous) section $s:X/i \rightarrow X$. Is this true? Any references ...
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Prove that two normed linear spaces are equivalent as metric spaces if and only if the norms are equivalent?

We have the two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on the vectorspace V. They're equivalent if there exists a $k>0$ and $K>0$ so that $k\|\cdot\|_a\le\|\cdot\|_b\le$ K$\|\cdot\|_a$ for all ...
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Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$. My workings so far: Because this is an if and only if proof we need to show it both ways. First let's assume ...
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Proving inequality with metrics

I was trying show that $|D(x,B) - D(y,B)| \le d(x,y)$ with $D(x,B) = \inf_{b \in B} d(x,b)$ and $(X,d)$ is a metric space. My try: $d(x,y) \ge d(x,b) -d(y,b) \ge \inf_{b\in B}d(x,B) - d(y,b)$ forall ...
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$\sup_{i \in I} \operatorname{dist}(x,S_i) \leq \operatorname{dist}(x, \bigcap_{i \in I}S_i)$

Prove that $$\sup_{i \in I} \operatorname{dist}(x,S_i) \leq \operatorname{dist}(x, \bigcap_{i \in I}S_i)$$ where $(X,d)$ is a metric space, $S_i \subseteq X : i \in I$, and $x \in X$. The question ...
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Why are the interior points in this question not the same?

I'm working on a question that wants me to write down the interior points of an interval contained in a metric space. $Let X=((1,7],d_{E})$ be a subspace of the metric space $(\mathbb{R},d_{E})$. Let ...
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Distance between two sets in a metric space in different conditions

let $(X,d)$ be a metric space and let $A,B\subseteq X$. we define the distance between $A$ and $B$ as: $$\operatorname{dist}(A,B)=\inf\{d(a,b):a \in A,b \in B\}$$ 1 show that for any $x \in X$, we ...
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Does this proof make sense and correct — is it written well enough?

I'm working on a tutorial question. The question asks whether the following claim is true or false, if it is true: one is supposed to provide a proof or counter-example otherwise if it's false. Let ...
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A question on Hamming metric/distance

Suppose $\sf{X}=\{0,1\}$, and $\sf{X}^n$ is the set of all binary sequences of length $n$. So the first question is that what does it mean by the convex closure of a subset $\sf{A}$ of $\{0,1\}^n$, ...
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Intersection of a Perfect and an open subsets of X

In Prof. George Bergman's Real analysis supplementary exercises, question 2.2:10. It is required to proof that the closure of the intersection of a perfect set $E$ with an open set $A$ is again ...
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A question on quasi-components

I have been doing some reading on general topology, connectedness in particular. Here is a question on a topological concept called quasi-component. Here is a definition: ...
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non compact nested closed sets in metric spaces.

Do you have an example for closed sets $...\subseteq F_4\subseteq F_3\subseteq F_2\subseteq F_1$ such that: $$\bigcap_{n=1}^\infty F_n=\emptyset $$ in $\mathbb{R}^n$ or a metric space?
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A natural-looking distance formula

The distance formula in one dimension is $$D_1 = |x_2-x_1|$$ From the Pythagorean theorem, the distance formula in two dimensions is $$D_2 = \sqrt{|x_2-x_1|^2 + |y_2-y_1|^2}$$ Now, in three ...
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Confused by an argument which is used in most triangle inequality proofs in metric spaces

I'm confused by the a proof of the triangle inequality. I was supposed to prove that a function is a metric, I proved everything else except the triangle inequality. Define $B(\mathbb{R})$ as the set ...
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A statement true about compacts but false about closed sets

Suppose that you have a metric space $X$. Could you give an example of a theorem or a statement that holds for compact sets but does not hold for closed sets? This question is motivated from a ...
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Is this proof correct and written in a understandable fashion?

Given a function $f(x,y) = \sqrt{(x_{1}-y_{1})^2 + (x_{2}-y_{2})^2 +...+(x_{n}-y_{n})^2}$ where $x,y$ $\in$ $\unicode{x211D}^n$, $f:\unicode{x211D}^n \times \unicode{x211D}^n \rightarrow ...
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$X$ metric separable then $C(X)$ separable

Is it true, that if $X$ is a separable metric space, then the space of all continuous functions on $X$ with the supremum metric is also separable?
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Clarification on the definition of the Lebesgue number of a metric space

Definition: Let $X$ be a metric space and $\mathcal{O}$ an open cover of $X$. A Lebesgue number for $\mathcal{O}$ is a positive number $\varepsilon$ with the property that every subset of $X$ of ...
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Is $\mathbf{R}^\omega$ in the uniform topology connected?

Let $\mathbf{R}^\omega$ be the set of all (infinite) sequences of real numbers. Then is this space connected in the uniform topology? How to determine this? The uniform metric $p \colon ...
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How to determine if this map is open or closed?

Given two supspaces $X:= [0,1]\cup[2,3]$ and $Y:=[0,2]$ of $\mathbf{R}$, let $f \colon X \to Y$ be defined as follows: $$f(x):= \left\{ \begin{array} {ll} x & \mbox{if $0\leq x\leq 1;$} \\ x-1 ...
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$d\left(\left(x_1,x_2\right),\left(y_1,y_2\right)\right)=|x_1-x_2|+|x_1-y_1|+|y_1-y_2|$ : complete?

Define the $\Bbb R^2$ metric $$ d\left( x,y \right) = \begin{cases} \left|x_2-y_2\right| &, x_1 = y_1 &&\text{(d1)}\\ \left|x_1-x_2\right|+\left|x_1-y_1\right|+\left|y_1-y_2\right| &, ...
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About Convergence of the Image of a Convergent Sequence Under a Uniformaly Convergent Sequence of Functions

Let $X$ be a topological space and $Y$ a metric space. Let $f_n \colon X \to Y$ be a sequence of continuous functions. Let $x_n$ be a sequence of points of $X$ converging to a point $x \in X$. Suppose ...
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Equivalent characterization of chain connectedness of a metric space

I'm having difficulty with proof. It is that the following is an equivalent characterization of chain connectedness for a metric space $M$: Point-wise boundedness at a point of an equicontinuous ...
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Infimum of distance in compact metric spaces.

Let $(A,d)$ be a metric space with $B\subseteq A$. If $B$ is compact, then it is bounded and closed. If $y\in A$ then there exists $x\in B$ so that $\inf\{d(y,z) : z\in B\} = d(y,x)$. It is reasonable ...
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Delta in continuity

Let $f: [a,b]\to\mathbb{R}$ be continuous, prove that it is uniform continuous. I know using compactness it is almost one liner, but I want to prove it without using compactness. However, I can use ...
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Sequences of functions

Suppose that $X$ is a compact metric space. Let: (a) $(f_n)$ be a sequence of real-valued continuous functions on $X$ (b) $(f_n)$ converges pointwise to a continuous function $f$ on $X$ (c) $f_n(x) ...