Hyperspaces of 0-dimensional spaces revisited

Miroslav M. Marjanović

An immediate reason for revisiting hyperspaces of $0$-dimensional spaces is the paper of Sh. Oka $[$Topology Appl. {\bf149} {\rm (2005), no. 1-3, 227--237]}, where the results of M. M. Marjanović $[$Publ. Inst. Math. $($Beograd$)$ $($N.S.$)$ {\bf14\,(28)} {\rm(1972), 97--109]} have been reproved. Aside from that, we take this opportunity to highlight the concepts of accumulation order and accumulation spectrum as a system of topological invariants which can be used efficiently in some situations of determining topological types in this class of spaces. In particular the Cartesian multiplication of the accumulation orders is an operation with respect to which the set of natural numbers $\mathbb{N}$ becomes a semi-group that can be used to reduce some subtle topological problems (for example, the existence of non-homeomorphic spaces with homeomorphic squares) to simple arithmetic verifications. The paper summarizes central ideas and details of the main constructions and may serve as an overview and introduction to this area of mathematics.