Small inductive dimension and Alexandroff topological spaces (Q2447134)

The authors study \(T_0\)-spaces that are Alexandroff, the latter meaning that every point, \(x\), has a minimal open neighbourhood, \(U_x\). Such spaces have a natural partial order, the specialization order, given by \(x\leq y\) iff \(x\in\overline{\{y\}}\) or, equivalently, \(y\in U_x\). The authors characterize and study the small inductive dimension of Alexandroff \(T_0\)-spaces in terms of the sizes of chains in this partial order, though without mentioning the order explicitly; this makes for somewhat cumbersome proofs.