Closed Intervals are Closed

Last updated Nov 6, 2022

Let $(X, \leq)$ be a Total Ordering equipped with the Order Topology. Then the following Sets are Closed:

1. $[a,b]$ for all $a, b \in X$ so $a \leq b$.
2. $(\leftarrow, a]$ and $[a, \rightarrow)$ for all $a \in X$
3. $X$ itself

(3) follows from the definition of a Topological Space $\checkmark$.

Note $(\leftarrow, a]^{C} = (a, \rightarrow)$ and $[a, \rightarrow)^{C} = (\leftarrow, a)$ for all $a \in X$. Because Open Rays generate the Order Topology, we know they are Open, showing that the Closed Rays are Closed and (2) follows $\checkmark$.

(1) follows after noting that for $a,b \in X$ so $a \leq b$, $[a,b] = (\leftarrow, b] \cap [a, \rightarrow)$ and Intersection of Closed Sets is Closed $\checkmark$. $\blacksquare$

• Closed Interval
• Closed Ray