Order Topology

Definition

An order topology is a specific topology that can be defined on any Totally Ordered Set.

A basis of the topology is defined as: $\mathcal{B}=\{(a,b),[a_0,b),(a,b_0]\subset X|a<b\}$ where $a_0$ and $b_0$ are the smallest and the largest element (if any) respectively.

Examples

${\mathbb{R}}^2$ with dictionary order: $\mathcal{B}=\{((a,b),(c,d))\subset {\mathbb{R}}^2|a<c\text{or}(a=c\text{and}b<d)\}$

Facts

A Topological Space defined by a Well-Ordered Set $X$ and an Order Topology on it is a normal space.

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