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.