Topology/Order

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


 * 1) $$(x\leq y)\land (y\leq x)\implies x=y$$ (antisymmetry)
 * 2) $$(x\leq y)\land (y\leq z)\implies x\leq z$$ (transitivity)
 * 3) $$(x\leq y)\lor (y\leq x)$$ (totality)

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

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