N ov 2 00 7 Cellular structures , quasisymmetric mappings , and spaces of homogeneous type

In this case, a set C ∈ C may be called a cell in X. A compact Hausdorff space with a cellular structure is a cellular space. For example, the usual construction of the Cantor set leads to a natural cellular structure, where the cells are the parts of the Cantor set in the closed intervals generated in the construction. Of course, a Hausdorff topological space with a base for its topology consisting of sets that are both open and closed is automatically totally disconnected, in the sense that there are no connected subsets with more than one element. The collection of all subsets of the space that are both open and closed is then an algebra of sets as well as a base for the topology. One can think of a cellular structure as a kind of geometric structure on such a space. For the sake of simplicity, let us restrict our attention to compact spaces, although one could consider non-compact spaces too. For instance, one might consider locally compact Hausdorff spaces that are σ-compact. Let (X, C) be a cellular space, and suppose that A ⊆ X is open and closed. In particular, A is compact, since X is compact. Because A is open and C is a base for the topology of X, A can be expressed as the union of