Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A\_{\alpha} : \alpha\in S\}$ such that $\overline{A}=\overline{\bigcup A\_{\alpha}}$ (where $\overline{A}$ denotes the closure of $A$). Can anyone find an example of a space which has the above property but is not zero-dimensional?



Zero-dimensional space
Any space which contains a dense set of isolated points will have the property. It is easy to construct (e.g. as subspaces of $\mathbb{R}^n$) such spaces with arbitrarily large dimension. For a one dimensional explicit construction see B.M.Scott's answer to this question: <https://math.stackexchange.com/questions/152390/cardinality-of-a-dense-open-set>.


