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 ← Subspaces Order Order Topology → 

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

  1. (x\leq y)\and (y\leq x)\implies x=y (antisymmetry)
  2. (x\leq y)\and (y\leq z)\implies x\leq z (transitivity)
  3. (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).


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

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