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:

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.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 \(\overline{S}\setminus S\).

created by Louis de Thanhoffer de Volcsey with thanks to Oliver Capon