Understanding the n-dimensional Simplex in Topology

The questions are (some background follows the questions):

1. Are 2- and 3-dim simplices really triangles and tetrahedrons (with lines connecting the vertices), or are they really just 3 and 4 sets of points.
2. If simplices only deal with numbers like $\mathbb{R}$ ("the standard n-simplex (or unit n-simplex) is the subset of $\mathbb{R}^{n+1}$"), or if there are simplices where the vectors can be arbitrary non-numerical data like in graph theory. Or is what I am thinking of CW-complexes, which seem to be a generalization of simplices.
3. If the topological ball is an example of simplices without numbers. Topologically a simplex is equivalent to a ball. A ball can be defined for a metric space, or even a topological space, which means it can be defined without numbers. These relate to CW-complexes.

The reasons for these questions are because I learned recently that a 0-dimensional simplex can be thought of as a vertex in a graph, and a 1-dimensional simplex can be thought of as an edge. An edge is essentially 2 vertices in the simplex. It is 2 things. But the thing is, an edge is a line connecting two vertices in graph theory, so in essence it is 3 things in graph theory: 2 vertices and 1 line. The line/edge can have it's own data or state. It seems the graph-theory vertices/edges are slightly different than the simplex defined vertices/edges. I understand why you can call a 1-dimensional simplex an edge, because it is essentially the data of two vertices (and that's it). But it's always drawn as a line connecting two points. Which got me wondering...

In terms of visualization, 2-dimensional simplices are drawn as triangles, and 3-dimensional as tetrahedron. Are they really triangles and tetrahedrons, or are they really just 3 and 4 sets of points. And for n-dimensional simplexes, they are n+1 collections of points. And in this way, they are like vectors, where each vector is a set of points. So simplexes are basically vectors (with I guess extra limitations about how they can connect). Wondering what I am missing. From wikipedia:

A simplex may be defined as the smallest convex set containing the given vertices.

A regular simplex is a simplex that is also a regular polytope.

Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is an n-dimensional manifold with corners.

And the mathematical definition of a simplex is:

$$C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for all }}i\right\}$$

This definition seems to say they are only about numbers, but maybe addition could be defined abstractly without numbers, which would agree with the abstract topological definition, not sure.

Since, geometrically, all points of a simplex are determined by convex combinations of the vertices, one can define a simplex as a list of vertices. For example, a triangle would be a triplet $(v_0,v_1,v_2)$ and in general, a $n$-simplex would be a $(n+1)$-tuple $(v_0,\dots, v_n)$.

As you might have noticed, the faces of an $n$-simplex (in a triangle: edges and vertices; in a tetrahedron: triangle faces, edges and vertices) are sublists of the simplex.

With this spirit one can define a simplicial complex, a set of simplices where every face of a simplex in the set is also in the set and every non-empmpty intersection o simplices in the set is also in the set. In a simplicial complex we can define a topology using the relation "being face of", which is basically the poset topology.

So you can make it the other way around, you can start with a set of abstract simplices (lists of vertices) and then you can realize it as a topological space. There is a canonical geometric realization, which consists of constructing the euclidean simplices by assigning coordinates in $\mathbb{R}^{n+1}$ to each vertex in a way that the vertices of any $k$-simplex ($k\leq n$) are affinely independent and constructing the polytopes with convex combinations. Here, we're giving the polytope the euclidean topology from $\mathbb{R}^{n+1}$.

To sum up:

1. Are 2-simplices and 3-simplices triangles and tetrahedrons? It depends on what definition you use. If you use the geometric definition, they are. If you use the abstract definition, it's their geometric realization. The important fact is that you can use both definitions indistinctly, since you can make the geometric thing from the abstract thing and viceversa.
2. Since you can forget about the numbers treating them as lists, it is not necessary to work on $\mathbb{R}^n$ unless you want to draw their geometric realization.
3. A topological ball is by definition homeomorphic to an euclidean ball, which is homeomorphic to the geometric realization of a simplex.

Added: there is also another common topology for simplicial complexes and CW-complexes, which is the weak topology.

I hope I've solved some of your questions.

• Ok cool, thank you! So a simplex can be defined without numbers, and is just vertices. That means there is an abstract polytope, yep, yay! – Lance Pollard May 4 '18 at 21:38
• Exactly! You're welcome – Javi May 4 '18 at 21:40
• What about the convex hull definition of a face. A convex hull is defined over $\mathbb{R}$ it looks like, so can convex hull be defined abstractly. – Lance Pollard May 5 '18 at 3:31
• That's what I called the "geometric definition". A face is just a subsimplex of dimension less than the whole simplex, so it's just taking a convex combination of some of the vertices of the original simplex. Hence, you can treat the faces as sublists of the whole list of vertices of the abstrac simplex. – Javi May 5 '18 at 11:09