  The objective of this paper is to examine the restriction of a right process
on a Radon topological space, excluding a negligible set, and investigate
whether the restricted object can induce a Markov process with desirable
properties. We address this question in three aspects: the induced process
necessitates only right continuity; it is a right process, and the
semi-Dirichlet form of the induced process is quasi-regular. The main findings
characterize the negligible set that meets the requirements within a
universally measurable framework. These characterizations can be employed to
generate instances of Markov processes that are non-right or (semi-)Dirichlet
forms that are non-quasi-regular. Specifically, we will construct an example of
a non-tight, strong Feller, symmetric right process on a non-Lusin Radon
topological space, whose Dirichlet form is not quasi-regular.
