# Tagged Questions

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 $$\mathcal{B}_f(c,r)=\{x \in a| ... 1answer 62 views ### Definition of disc and open ball I have the following definitions in my notes for arbitrary discs and open balls -$$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$The ... 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 = ... 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 ... 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, ... 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 ... 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 ... 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} ... 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. ... 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. 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 ... 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 ... 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 ... 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)\}, ...
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 ...