Topology/Order

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

1. $(x\le y)\wedge (y\le x)⟹x=y$ (antisymmetry)
2. $(x\le y)\wedge (y\le z)⟹x\le z$ (transitivity)
3. $(x\le y)\vee (y\le x)$ (totality)

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

Definition

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

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