  In the theory of zero-dimensional systems and their relation to
$C^*$-algebras, Poon (1990) introduced a class of closed sets. We call the
closed sets quasi-sections. Medynets (2006) introduced basic sets that are part
of quasi-sections in his study of aperiodic zero-dimensional systems and their
relation to Bratteli--Vershik models and $C^*$-algebras. Downarowicz and Karpel
(2019) introduced the notion of decisiveness in the theory of Bratteli--Vershik
models. We previously clarified that particular quasi-sections can be the
"bases" of the decisive Bratteli--Vershik models for zero-dimensional systems
with dense aperiodic orbits. We call them continuously decisive quasi-sections.
However, even the basic topological properties of quasi-sections and the basic
sets have not been studied systematically. This paper presents such a
systematic study. Some properties are defined, stated, and proved in the
general settings of compact Hausdorff topological dynamics. For example, if a
topological dynamical system has dense aperiodic orbits and no wandering
points, then every basic set is continuously decisive. If a zero-dimensional
system has dense aperiodic orbits, then there exists a minimal continuously
decisive basic set such that for every minimal set, there exists a unique
common point.
