The property that you are describing is called *coabsoluteness*. In other words, two regular spaces are said to be coabsolute if their regular open algebras are isomorphic. In the paper, A Characterization of Coabsoluteness for a Class of Metric Spaces by Catherine Gates, theorem 2.3 says that two locally compact metric spaces $X$ and $Y$ are coabsolute if and only if $d(X)=d(Y)$. Here $d(X)$ denotes the density of $X$, i.e. the $d(X)$ is smallest cardinal such that there is a dense subset of $X$ of cardinality $d(X)$. In particular, the open interval, closed inverval, $S^{1}$, $S^{2}$ and the Hawaiian earring are all locally compact separable metric spaces, so they are all coabsolute. Therefore these spaces all have isomorphic regular open algebras.