simplex and power set

I read the following:

Let $M$ be a set. The simplex on $M$ is the set of all subsets of $M$; we denote this by $\Delta_M$. We will sometimes refer to the elements of $M$ as vertices of $\Delta_M$. A simplicial complex on $M$ is a subset of $\Delta_M$ which is closed under the taking of subsets. If $\Gamma$ is a simplicial complex on $M$ and $F\in \Gamma$, we say that $F$ is a face of $\Gamma$. We require that simplicial complexes be nonempty; that is, the empty set must always be a face.

Is $\Delta_M$ what we call the full simplex? and isn't this simply $2^M$ the power set of $M$? What do we mean by closed under the taking, do we mean closed under union operation? is this what we call an abstract simplicial complex and how does this relate to the geometric notion of simplicial complex?

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1 Answer

That definition seems to assume that $M$ is a finite set, since the members of a simplicial complex are normally required to be finite subsets of the underlying set. Yes, $\Delta_M$ is the power set of $M$. If $\Gamma\subseteq\Delta_M$, $\Gamma$ is closed under the taking of subsets if and only if $X\subseteq Y\in\Gamma$ implies that $X\in\Gamma$.

More generally, if $\mathscr{A}$ is any collection of sets, we say that $\mathscr{A}$ is closed under the taking of subsets (or simply closed under taking subsets) if every subset of a member of $\mathscr{A}$ is also a member of $\mathscr{A}$. Yet another way to say this is that for each $A\in\mathscr{A}$, $\wp(A)\subset\mathscr{A}$. Unions are not involved here.

If $K$ is a geometric simplicial complex, let $M$ be the set of its vertices; we’ll build an associated abstract simplicial complex $\Gamma$ on $M$. Each $d$-dimensional face of $K$ has $d+1$ vertices; let that set of $d+1$ vertices belong to $\Gamma$. For example, if $K$ consists of a tetrahedron with vertices $v_1,v_2,v_3$, and $v_4$, a triangle with vertices $v_1,v_2$, and $v_5$, and a segment with vertices $v_5$ and $v_6$, $M=\{v_1,v_2,v_3,v_4,v_5,v_6\}$, and

\begin{align*}\Gamma&=\Big\{\{v_1,v_2,v_3,v_4\},\{v_1,v_2,v_3\},\{v_1,v_2,v_4\},\{v_1,v_3,v_4\},\{v_2,v_3,v_4\},\\ &\quad\;\;\,\{v_1,v_2\},\{v_1,v_3\},\{v_1,v_4\},\{v_2,v_3\},\{v_2,v_4\},\{v_3,v_4\},\{v_1\},\{v_2\},\\ &\quad\;\;\,\{v_3\},\{v_4\},\varnothing,\{v_1,v_2,v_5\},\{v_1,v_5\},\{v_2,v_5\},\{v_5\},\{v_5,v_6\},\{v_6\}\Big\}\;. \end{align*}

Going in the other direction, I’ll quote Wikipedia:

If $K$ is [a] finite [abstract simplicial complex], then we can describe $|K|$ more simply. Choose an embedding of the vertex set of $K$ as an affinely independent subset of some Euclidean space $\Bbb R^N$ of sufficiently high dimension $N$. Then any face $X \in K$ can be identified with the geometric simplex in $\Bbb R^N$ spanned by the corresponding embedded vertices. Take $|K|$ to be the union of all such simplices.

(Here $|K|$ is the geometric realization of $K$.)

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you mean $X\subseteq Y\in \Gamma$ implies $X\in \Gamma$. – palio Apr 12 '12 at 11:38
@palio: Fixed; thanks! – Brian M. Scott Apr 12 '12 at 11:40