In the book Elements of Algebraic Topology by James R. Munkres, a simplicial complex in $\mathbb{R}^N$ is defined as follows:
A simplicial complex $K$ in $\mathbb{R}^N$ is a collection of
simplices in $\mathbb{R}^N$ such that:
- Every face of a simplex of $K$ is in $K$.
- The intersection of any two simplexes of $K$ is a face of each of them.
A face of a $n$-simplex $\sigma$ spanned by $a_0, \ldots a_n$ is defined as follows:
Any simplex spanned by a subset of $\{ a_0, \ldots, a_n \}$ is called
a face of $\sigma$.
Moreover:
The points $a_0, \ldots, a_n$ that span $\sigma$ are called the
vertices of $\sigma$; the number $n$ is called the dimension of
$\sigma$.
According to this definition it seems to me that a face cannot be empty (what would be $n$ in that case?), which would imply that the intersection of two simplexes of a simplicial complex cannot be empty. Nevertheless, later on in this book, some examples of simplicial complexes seem to have simplexes with an empty intersection, so I am confused.
Best Answer
This is in fact a delicate question. Munkres defines a simplex as the convex hull of a set of $n+1$ geometrically independent vertices $\{a_0,\dots,a_n\}$, but does not say whether he requires $n \ge 0$ or formally allows $n = -1$ in which case $n+1= 0$, i.e. $\{a_0,\dots,a_n\}= \emptyset$.
Allowing $n = -1$ , one could regard the empty set as a simplex having $0$ vertices: As its vertex set take $\emptyset$. It has $0$ elements and its points are geometrically independent (simply because there are no such points). The convex hull of $\emptyset$ is again $\emptyset$.
However, this would be a non-standard approach, usually one does not consider empty simplices. See for example here. In my opinion it would be better to replace 2. by
But allowing the empty simplex has an interesting effect: The chain groups $C_p(K)$ of $K$ are introduced in §5, and for $p=-1$ it would then have one generator (represented by the empty simplex). This produces nothing else than the augmented chain complex introduced in §13.