The terminology used in this chapter follows that of Bourbaki [18]. However, the terms “scattered”, “totally disconnected”, and “totally separated” have sometimes different meanings in the literature. For example, spaces that are called “scattered” in the present book are called “zero-dimensional” in [102], while a “scattered” space in [102] is a topological space in which every non-empty subset admits an isolated point.

Table 2.1 Summary Table (*X* non-empty)

The Cantor ternary set was described by Cantor in [21, note 11 p. 46]. It can be shown that every totally disconnected compact metrizable space that is perfect is homeomorphic to the Cantor set (see for example [48, Corollary 2–98]).

A non-empty topological space *X* is scattered if and only if \({{\mathrm{ind}}}(X) = 0\) (see the Notes on Chap. 1, p.19, for the definition of the small inductive dimension \({{\mathrm{ind}}}(X)\)). The question of the existence of scattered metrizable spaces with positive topological dimension remained open for many years (cf. [18, note 1 p. IX.119]). An affirmative answer to this question was finally given by Roy [96, 98] who constructed a scattered metrizable space *X* with \(\dim (X) = 1\).

The notion of a totally disconnected space and that of a totally separated space were respectively introduced by Hausdorff [47] and by Sierpinski [99]. In [99], Sierpinski described a totally disconnected subset of \(\mathbb {R}^2\) that is not totally separated and a totally separated subset of \(\mathbb {R}^2\) with positive topological dimension.