Empty set in a simplicial complex Should the empty set be considered a simplex in a simplicial complex?
Which justifications exist for the answer?
I guess it is somewhat comparable to $1$ not being a prime number.
 A: An abstract simplicial complex is a set $V$ of vertices together with a set $F$ of faces (which you seem to refer to as simplices) such that $A\subseteq B$ and $B\in F$ implies $A\in F$ (closed under inclusion). So if $F$ is nonempty then it also contains $\emptyset$ as an element. It means the empty simplex is a face.
Some other answers are addressing the question of the existence of an empty simplex understood as a simplicial complex. That also exists, it is the simplicial complex with $V=\emptyset$ and $F=\{\emptyset\}$. It is actually super useful. It should be thought of as the $(-1)$-dimensional sphere, since its suspension is the $0$-dimensional sphere. Its geometric realization is the empty topological space (with no points and the empty set as the only open set).
A: This is a useful construction in homotopy theory and is probably easiest explained as the augmented simplex category.
A: I think your answer is yes.  If the intersection of two simplices in a simplicial complex is a face (sub-simplex) of each, then an empty set must be a simplex.
