  In this paper, we will investigate how radiality occurs in topological spaces
by considering neighbourhood bases generated by nests. We will define a new
subclass of radial spaces, called the independently-based spaces, that contains
LOTS, GO-spaces and spaces with well-ordered neighbourhood bases. We show that
first-countable spaces are precisely the independently-based, strongly
Fr\'echet spaces and we give an example of a Fr\'echet-Urysohn space that is
neither independently-based nor strongly Fr\'echet.
