Denoting tolerable range for common values in two sequences

Question

Given set X and set Y, common values can be denoted in a new set via set intersection, so: Z = X ∩ Y. However, let's say set X = {1,2,3,4} and set Y = {1.3, 2.3, 2.7 4.3}. All values in set X are within .3 of some value in set Y, yet Z = X ∩ Y would be empty since the values aren't exactly the same. How can you specify that set Z includes the intersection of values in X and Y, within the tolerable range of plus or minus .3?

Answer 1

You can define the so-called $0.3$ neighborhood $A^{(0.3)}$ of $A$ as the set of all numbers $x$ such that $|x-a| \leq 0.3$ for some $a \in A$. Then the property you have is $ B \subseteq A^{(0.3)}$.

General definition: $\epsilon$ neighborhood $A^{(\epsilon)}$ of a set $A$ is the set of all numbers $x$ such that $d(x,a) \leq \epsilon$ for some $a \in A$. [ $(d(x,a)$ is the distance between $x$ and $a$]. This definition makes sense in any metric space.