In wikipedia, I see that
A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover by disjoint open sets.
A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.
The three notions above agree for separable, metrisable spaces.
However there is no reference given. Does anybody know a book or notes that have a proof of the equivalence of the three conditions in the separable, metrizable case? It is obvious that (1) implies (2) in general, but the implications (2) to (3) and (3) to (1) are not straightforward, I think.
Also, any reference on books for the general theory of zero-dimensional spaces in the above senses would be appreciated.
Best Answer
(2) implies (3) is straightforward, in any T_1 space. In fact you that only need that any two member open covering has a disjoint open refinement. Suppose $x \in U$ and $U$ is open in $X$. Then $\{ U, X \setminus \{ x \} \}$ is such a covering and the member of the refinement containing $x$ will be a clopen nbhd. within $U$.
(3) implies (1) can be proved in any Lindelof space, which includes separable metrisable spaces. Given an open covering, refine it with basic clopen sets and take a countable subcovering of that, say $\{ C_n | n \in \mathbb N \}$. Let $D_1 = C_1$ and each $D_{n+1} = C_{n+1} \setminus (C_1 \cup ... \cup C_n \}$. $\{D_n | n \in \mathbb N \}$ is a locally-finite, clopen refinement. The union and intersection of any locally finite collection of clopen sets are clopen, so the partition derived from $\{ D_n | n \in \mathbb N \}$ is a refinement of the original covering into disjoint clopen sets.