|
For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (-∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (-∞, ∞) = X. The totally ordered set X turns into a topological space if we define a subset to be open if and only if it is a (possibly infinite) union of such open intervals. This is called the order topology on X; it is always normal?. Unless otherwise stated, it is understood that this topology is being used on a totally ordered set. |
|
For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (-∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (-∞, ∞) = X. The totally ordered set X turns into a topological space if we define a subset to be open if and only if it is a (possibly infinite) union of such open intervals. This is called the order topology on X; it is always a normal Hausdorff space. Unless otherwise stated, it is understood that this topology is being used on a totally ordered set. |
![[Home]](wikipedia.png)