General Topology - Order Topology

The Order Topology

Definition 8. [Order Topology]

Let $X$ be an ordered set. Then, $\mathfrak{B}$ is the collection of the following subsets:

1. $(a, b)$, with $a, b \in X$ and $a < b$;
2. $[a_0, b)$, where $b \in X$ and $a_0$ is the smallest element of $X$;
3. $(a, b_0]$, where $a \in X$ and $a_0$ is the largest element of $X$.

$\mathfrak{B}$ generates a topology $\mathfrak{T}$, which is called Order Topology.

Definition 9. [Rays]

Let $X$ be a set with simple order relation, and $a \in X$. Then, the following $4$ kinds of subsets are called rays determined by $a$:

1. $(a, +\infty)$, also called open rays;
2. $(-\infty, a)$, also called open rays;
3. $[a, +\infty)$, also called closed rays;
4. $(-\infty, a]$, also called closed rays;