Non-trivial (regular) open semigroups of the open unit interval $(0,1)$?

Question

This is a follow-up question to the one I asked here.

Namely, are there examples of open (or even regular open) subsets of $(0,1)$ which are multiplicative subsemigroups of $(0,1)$ but are not of the form $(0,a)$ for $a\in (0,1)$?

Answer 1

The map $x\mapsto -\log x$ is an isomorphism of topological semigroups from $((0,1),\cdot)$ to $((0,\infty),+)$, so you are equivalently asking about (regular) open additive subsemigroups of $(0,\infty)$ that are not of the form $(a,\infty)$. A simple example is $(1,1.5)\cup(2,\infty)$.

More generally, the subsemigroup of $(0,\infty)$ generated by an open interval $(a,b)$ will be the union $\bigcup_{n\in\mathbb{Z}_+}(na,nb)$. If $2a\geq b$, then this union will not be just a single interval. However, it will always contain an interval of the form $(c,\infty)$, since there exists $n$ such that $(n+1)a<nb$ and then all of the following intervals will overlap so that all of $(na,\infty)$ is contained in the union. So, any nonempty open subsemigroup of $(0,\infty)$ contains an interval of the form $(c,\infty)$.