Topology/Order

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

$(x\leq y)\and (y\leq x)\implies x=y$ (antisymmetry)
$(x\leq y)\and (y\leq z)\implies x\leq z$ (transitivity)
$(x\leq y)\or (y\leq x)$ (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 $(X,\leq)$ .

[edit] 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 $X$

Topology/Order

From Wikibooks, the open-content textbooks collection

[edit] Definition

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