1
vote
2answers
89 views

Why in open balls is radius $r>0$?

the usual definition is the following: Def.1: let be $(a,f)$ a metric space, $c \in a$ and $r \in \Bbb{R}_{>0}$, the open ball of radius $r$ about $c$ is the set $$\mathcal{B}_f(c,r)=\{x \in a| ...
2
votes
1answer
62 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
0
votes
0answers
30 views

Notation problem with a set of tuples and a metric

The first question: Assume we have tuples $T_i = (x_i, \vec{c}_i)$ ($x_i$ is the name of the object which is characterized by $\vec{c}_i$ in a d-dimensional space) and define a set of them $TS = ...
5
votes
3answers
100 views

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 ...
3
votes
2answers
63 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
1
vote
1answer
59 views

Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
1
vote
1answer
78 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
2
votes
4answers
80 views

Definition of open set/metric space

On Proof Wiki, the definition of an open set is stated as Let $(M,d)$ be a metric space and let $U\subset M$, then $U$ is open iff for all $y\in U$, there exists $\epsilon \in \mathbb{R}_{>0}$ ...
5
votes
2answers
126 views

Intuitive explanation of ball-based definition for continuity of functions in metric spaces

First of all, hat tip to @Fayz for providing this definition. Backstory: I broke my glasses several days ago and, in the meantime, this important definition was written on a board I could not see. ...
2
votes
2answers
333 views

two notation: semi-metric and pesudometric

There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different? Thanks ahead.
1
vote
4answers
164 views

Definition of a metric

I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along ...
1
vote
1answer
100 views

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 ...
8
votes
4answers
429 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
5
votes
2answers
635 views

On the definition of the Hausdorff distance

$\newcommand{\dist}{\mathrm{dist}\,}$ Let $M$ be a metric space and $\emptyset\neq A,B\subset M$ bounded closed subsets. The Hausdorff distance is defined as $$h(A,B)=\max\{\dist(A,B),\dist(B,A)\},$$ ...
3
votes
3answers
1k views

Definition of open and closed sets for metric spaces

For metric spaces, the definition of an open set $U\subset X$ is that it is a set which for any point $u\in U$ in the set there exists some $\epsilon>0$ such that the open ball ...