I'm trying to prove the following: We have a DLO without endpoints M, and a group operation on M, which is continuous in the interval topology. I want to prove: if $b<c$ then for every $a \in M$ we have that $b+a<c+a$.
I'm trying to consider the supremum, defined for a fixed $a \in M$: $s:=sup \{ x| x+a<b+a \}$, and show that we must have $s=b$, but since this is an arbitrary DLO I can't have $s \in M$ so I'm a bit stuck. any ideas?
Thanks!