Topology/Order

Recall that a set ${\displaystyle X}$ is said to be totally ordered if there exists a relation ${\displaystyle \leq }$ satisfying for all ${\displaystyle x,y,z\in X}$

1. ${\displaystyle (x\leq y)\land (y\leq x)\implies x=y}$ (antisymmetry)
2. ${\displaystyle (x\leq y)\land (y\leq z)\implies x\leq z}$ (transitivity)
3. ${\displaystyle (x\leq y)\lor (y\leq x)}$ (totality)

The usual topology ${\displaystyle {\mathcal {U}}}$ on ${\displaystyle \mathbb {R} }$ is defined so that the open intervals ${\displaystyle (a,b)}$ for ${\displaystyle a,b\in \mathbb {R} }$ form a base for ${\displaystyle {\mathcal {U}}}$. It turns out that this construction can be generalized to any totally ordered set ${\displaystyle (X,\leq )}$.

Let ${\displaystyle (X,\leq )}$ be a totally ordered set. The topology ${\displaystyle {\mathcal {T}}}$ on ${\displaystyle X}$ generated by sets of the form ${\displaystyle (-\infty ,a)}$ or ${\displaystyle (a,\infty )}$ is called the order topology on ${\displaystyle X}$