  The Contextuality-by-Default approach to determining and measuring the
(non)contextuality of a system of random variables requires that every random
variable in the system be represented by an equivalent set of dichotomous
random variables. In this paper we present general principles that justify the
use of dichotomizations and determine their choice. The main idea in choosing
dichotomizations is that if the set of possible values of a random variable is
endowed with a pre-topology (V-space), then the allowable dichotomizations
split the space of possible values into two linked subsets ("linkednes" being a
weak form of pre-topological connectedness). We primarily focus on two types of
random variables most often encountered in practice: categorical and
real-valued ones (including continuous random variables, greatly
underrepresented in the contextuality literature). A categorical variable (one
with a finite number of unordered values) is represented by all of its possible
dichotomizations. If the values of a random variable are real numbers, then
they are dichotomized by intervals above and below a variable cut point.
