In the book "Algebraic Topology" by E Spanier, a simplicial complex is define a follows -

D1 - A simplicial complex $K$ consists of a set $\{v\}$ of vertices and a set $\{s\}$ of non empty subsets of $\{v\}$ called simplexes such that

a) a set consisting of a single vertex is a simplex.

b) any non empty subset of a simplex is a simplex.

In the book "An Introduction to Algebraic topology" by J J Rotman it is defined as follows -

D2 - A finite simplicial complex $K$ is a finite collection of simplexes in some Euclidean space such that -

a) if $s\in K$ then every face of $s$ also is in $K$

b) if $s,t \in K$ then $s \cap t$ is either empty or a common face of $s$ and $t$.

Here a simplex $s$ is a convex combination of a set of points (called vertices) $Vert(s)=\{v_0 ,...,v_q\} \subset \mathbb{R}^n$ that are affine independent. A face of $s$ is a simplex $s'$ such that $Vert(s') \subset Vert(s)$.

My question is how are these two definitions the same (if they are at all)?

D2 seems more geometric while D1 more abstract. Of course I am assuming that in D1 I can impose finiteness on the definition by requiring the vertex set $\{v\}$ to be finite (or can't I?) and thus take the vertex set to be a subset of $\mathbb {R}^n$ as is done in D2.



If you have a complex from definition D2 you can go to D1 where the set of vertices is just the set $V$. Simplices are just subsets of $V$. A face of a simplex $\tau$ is just another simplex $\sigma$ spanned by some nonempty subset of its vertices. Of course every nonempty subset of vertices spans a simplex. So this why you have condition b) in D1. Similarly with a).

There is just easy to see that a simplex is completely determinated by its set of vertices.

However these two definition are not equivalent. They are equivalent under assumption in D1 that the set $V$ is finite. There are infinite complexes as well.

The difference between these two definition is that in the second one you define a subspace of some $\mathbb R^n$ but in the first one you have information how the simplices are glued together. Which one is a face of another.

For an abstract simplicial complex there are infinitely many of its realization in $\mathbb R^n$ for some appropriate $n$ (if complex is finite). But all of them are isomorphic.


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