Recall that an abstract simplicial complex consists of a family of finite sets $K\subseteq 2^V$ such that $\sigma\in K$ and $\tau\subseteq\sigma$ implies $\tau\in K$. (Sometimes it is also assumed that $\{v\}\in K$ for all $v\in V$.) Elements of $K$ are called faces of $K$. Dimension of a face is its cardinality minus 1.

According to the above definition, if $S\not=\emptyset$, then the empty set $\emptyset$ is a face of the complex (of dimension $-1$, since it has $0$ elements). Applying the usual definition of simplicial homology gives us what is called reduced homology, which is often much better-behaved than the nonreduced one (obtained when we forget about the empty face).

In this context it is important to distinguish between the empty simplicial complex $K_e=\{\emptyset\}$ and the void simplicial complex $K_v=\{\}$, which may be understood as the $(-1)$-sphere and the $(-1)$-disk. (In particular $H_{-1}(K_e)=\mathbb{Z}$, $H_n(K_e)=0$ for all $n\not=0$, and $H_m(K_v)=0$ for all $m$.)

Recall that an abstract simplicial complex consists of a family of finite sets $K\subseteq 2^V$ such that $\sigma\in K$ and $\tau\subseteq\sigma$ implies $\tau\in K$. (Sometimes it is also assumed that $\{v\}\in K$ for all $v\in V$.) Elements of $K$ are called faces of $K$. Dimension of a face is its cardinality minus 1.

According to the above definition, if $S\not=\emptyset$, then the empty set $\emptyset$ is a face of the complex (of dimension $-1$, since it has $0$ elements). Applying the usual definition of simplicial homology gives us what is called reduced homology, which is often much better-behaved than the nonreduced one (obtained when we forget about the empty face).

In this context it is important to distinguish between the empty simplicial complex $K_e=\{\emptyset\}$ and the void simplicial complex $K_v=\{\}$, which may be understood as the $(-1)$-sphere and the $(-1)$-disk. (In particular $H_{-1}(K_e)=\mathbb{Z}$, $H_n(K_e)=0$ for all $n\not=-1$, and $H_m(K_v)=0$ for all $m$.)

Recall that an abstract simplicial complex consists of a family of finite sets $K\subseteq 2^V$ such that $\sigma\in K$ and $\tau\subseteq\sigma$ implies $\tau\in K$. (Sometimes it is also assumed that $\{v\}\in K$ for all $v\in V$.) Elements of $K$ are called faces of $K$. Dimension of a face is its cardinality plus 1.

According to the above definition, if $S\not=\emptyset$, then the empty set $\emptyset$ is a face of the complex (of dimension $-1$, since it has $0$ elements). Applying the usual definition of simplicial homology gives us what is called reduced homology, which is often much better-behaved than the nonreduced one (obtained when we forget about the empty face).

In this context it is improtant to distinguish between the empty simplicial complex $K_e=\{\emptyset\}$ and the void simplicial complex $K_v=\{\}$, which may be understood as the $(-1)$-sphere and the $(-1)$-disk. (In particular $H_{-1}(K_e)=\mathbb{Z}$, $H_n(K_e)=0$ for all $n\not=0$, and $H_m(K_v)=0$ for all $m$.)

Recall that an abstract simplicial complex consists of a family of finite sets $K\subseteq 2^V$ such that $\sigma\in K$ and $\tau\subseteq\sigma$ implies $\tau\in K$. (Sometimes it is also assumed that $\{v\}\in K$ for all $v\in V$.) Elements of $K$ are called faces of $K$. Dimension of a face is its cardinality minus 1.

According to the above definition, if $S\not=\emptyset$, then the empty set $\emptyset$ is a face of the complex (of dimension $-1$, since it has $0$ elements). Applying the usual definition of simplicial homology gives us what is called reduced homology, which is often much better-behaved than the nonreduced one (obtained when we forget about the empty face).

In this context it is important to distinguish between the empty simplicial complex $K_e=\{\emptyset\}$ and the void simplicial complex $K_v=\{\}$, which may be understood as the $(-1)$-sphere and the $(-1)$-disk. (In particular $H_{-1}(K_e)=\mathbb{Z}$, $H_n(K_e)=0$ for all $n\not=0$, and $H_m(K_v)=0$ for all $m$.)