No. Let A = {a} (a singleton set) then P(A) = {a, 0}where 0 is the null (empty) set.
Speaking in a grammatical sense, one sets a precedent, not a precedence.
when the sun/moon appears to slowly meet the horizon and disappear
An empty set is considered a finite set because it contains zero (0) elements and zero is a finite number.
That is the definition of a closed set.