User:Dfeuer/Interior Point of Interval/Densely Linearly Ordered Space/Lemma
Jump to navigation
Jump to search
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} }$