In this 6th week of class, we began by a Q&A session in order to prepare for the midterm. We then went on to lay the groundwork for our next big property of graphs: planarity.
The idea of planarity is simple enough:
can you draw a graph in the plane without two edges crossing each other?
The precise mathematical formulation iof this question s a little tricky however and requires some notions from topology:
A set \(U\subset mathbb{R}^n\) is open if for each \(x\in U\) there exist some value \(\epsilon>0\) such that the ball \(y \in \mathbb{R}^n\vert \vert \vert y-x\vert \vert <\epsilon\) lies in \(U\).
The closure of a set \(S\in \mathbb{R}^n\) consists of all points \(x \in \mathbb{R}^n\) such that there exists a sequence \((x_i)_i \subset S\) with \(\lim x_i= x\). We denote the closure by \(\overline{S}\). The boundary of a set is given by \(\overline{S}\setminus S\).
I gave examples in \(\mathbb{R}\) of these concepts: \(]a,b[\) is an open set, \([a,b]\) is closed and \([a,b]\) is neither open nor closed.