F-9 - Topological Characterizations of Separable Metrizable Zero-Dimensional Spaces

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A space X is called zero-dimensional if it is nonempty and has a base consisting of clopen sets, that is, if for every point x ∈X and for every neighborhood U of x there exists a clopen subset C ⊆X such that x ∈C ⊆U. A nonempty subspace of a zero-dimensional space is again zero-dimensional and products of zero-dimensional spaces are zero-dimensional. A space X is totally disconnected if for all distinct points x, y in X there exists a clopen set C in X such that x ∈C but y ∉C. Every zero-dimensional space is totally disconnected. The question as to whether every totally disconnected space is zero-dimensional, naturally arises. If this were true, then checking whether a given space is zero-dimensional would be simpler. However, the answer to this question is the negative. In the first part of this chapter, interest is focused on theorems that state nontrivial and useful topological characterizations of zero-dimensional separable metrizable spaces. This chapter briefly mentions a few results for general Tychonoff spaces.