Reverse Ordering generates same Topology

Last updated Nov 1, 2022

Let $(X, \leq)$ be a Total Ordering with the Order Topology. Let $\leq’$ be the Reverse Ordering on $X$. Then $(X, \leq’)$ has the same Order Topology as $(X, \leq)$.

We must simply check that they are generated by the same collection. Indeed for any $c \in X$, we have $(c, \rightarrow){\leq} = ( \leftarrow, c){\leq’}$ and $( \leftarrow, c){\leq} = (c, \rightarrow){\leq’}$. $\blacksquare$

1. Another way to state this is the Identity Function is a Homeomorphism between a Total Ordering and its Reverse Ordering.