  Based on topological Rudin's Lemma, we investigate two new kinds of sets -
Rudin sets and well-filtered determined sets in $T_0$ topological spaces. Using
such sets, we formulate and prove some new characterizations for well-filtered
spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work
on well-filtered spaces. Our study also lead to a new class of spaces - strong
$d$-spaces and some problems whose solutions will strengthen our understanding
of the related structures.
