Editing Abstract polytope (section)

===Least and greatest faces===
Just as the number zero is necessary in mathematics, so also every set has the [[empty set]] ∅ as a subset. In an abstract polytope ∅ is by convention identified as the ''least'' or ''null'' face and is a subface of all the others.{{why|date=July 2020|reason=Why is the empty set made a member? Many other subsets are not members.}} Since the least face is one level below the vertices or 0-faces, its rank is −1 and it may be denoted as ''F''<sub>−1</sub>. Thus F<sub>−1</sub> ≡ ∅ and the abstract polytope also contains the empty set as an element.<ref>{{Harvnb |McMullen |Schulte |2002 |loc=}}</ref> It is usually not realized, though the lack of its realization could be interpreted as it being realized as the set containing no points, the empty set.

There is also a single face of which all the others are subfaces. This is called the ''greatest'' face. In an ''n''-dimensional polytope, the greatest face has rank = ''n'' and may be denoted as ''F''<sub>''n''</sub>. It is sometimes realized as the interior of the geometric figure.

These least and greatest faces are sometimes called ''improper'' faces, with all others being ''proper'' faces.<ref name="ARP23"/>