User:Dfeuer/Interior Point of Interval/Densely Linearly Ordered Space/Lemma

Theorem
Let $\left({X, \le, \tau}\right)$ be a close packed and linearly ordered space.

Let $a, b \in X$.

Let $I = \left[{{a}\,.\,.\,{b}}\right] = \left\{{ x \in X: a \le x \le b }\right\}$.

Then the interior of $I$ is $\left({{a}\,.\,.\,{b}}\right) \cup M_a \cup M_b$, where:


 * $M_a = \cases {

\left\{{a}\right\} & \text{if $a$ is a lower bound for $X$} \\ \varnothing & \text{otherwise} }$


 * $M_b = \cases {

\left\{{b}\right\} & \text{if $b$ is an upper bound for $X$} \\ \varnothing & \text{otherwise} }$