Characterization of some zero-dimensional separable metric spaces

Abstract

Let X 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 Y which can be written as an increasing union ∪ n = 1 ∞ F n \cup _{n = 1}^\infty {F_n} of closed sets so that for all n ∈ N n \in {\mathbf {N}} , F n {F_n} is a copy of X X which is nowhere dense in F n + 1 {F_{n + 1}} . If moreover X X contains a closed nowhere dense copy of itself, then Y Y is homeomorphic to Q × X {\mathbf {Q}} \times X where Q {\mathbf {Q}} denotes the space of rational numbers. This gives us topological characterizations of spaces such as Q × C {\mathbf {Q}} \times {\mathbf {C}} and Q × P {\mathbf {Q}} \times {\mathbf {P}} .