Let $a<b$, can we have a form of $(a,b)$ that is closed in $S\subset\mathbb{R}$
I know that $(a,b)$ itself is being closed in $S=(a,b)$.
and $(a,b)$ is clopen in $S'=(a,b)\cup(b+1,b+2)$
This is kind of cheating to my question, except above examples I want some proper subset $(a,b)\subset S\subset \mathbb{R}$ that is being closed.
Thanks for any comments