Characterization of some zero-dimensional separable metric spaces

Abstract:

Let $X$ be a separable metric zero-dimensional space for which all nonempty clopen subsets are homeomorphic. We show that, up to homeomorphism, there is at most one space $Y$ which can be written as an increasing union $\cup _{n = 1}^\infty {F_n}$ of closed sets so that for all $n \in {\mathbf {N}}$, ${F_n}$ is a copy of $X$ which is nowhere dense in ${F_{n + 1}}$. If moreover $X$ contains a closed nowhere dense copy of itself, then $Y$ is homeomorphic to ${\mathbf {Q}} \times X$ where ${\mathbf {Q}}$ denotes the space of rational numbers. This gives us topological characterizations of spaces such as ${\mathbf {Q}} \times {\mathbf {C}}$ and ${\mathbf {Q}} \times {\mathbf {P}}$.