TITLE: Parabola - Definition as a locus of points
QUESTION [3 upvotes]: On Wikipedia, a parabola is defined as follows:

A parabola is a set of points such that, for any point $P$ of the parabola, the distance $|\overline{PF}|$ to a fixed point $F$, the focus, is equal to the distance $|\overline{Pl}|$ to a fixed line $l$ 

Isn't this definition wrong since an empty set is a parabola according to it?
Is the following definition correct?

For any set of points $S$, $S$ is a parabola if and only if
  there exists a point $F$ and a line $l$ such that
  $F$ isn't on $l$ and for any point $P$, $P$ is in $S$ if and only if $|\overline{PF}|=|\overline{Pl}|$.

REPLY [3 votes]: Your criticism is justified. The definition should say, not that the parabola is a set of points such that…, but that it is the set of all points such that… (in other words, it's the locus of those points). On the other hand, that article says that “The focus does not lie on the directrix”.
{"set_name": "stack_exchange", "score": 3, "question_id": 3235419}