Argthtjtr

For $a_0,...,a_k$ affinely independent points in $mathbb{R}^N$ for $Nge k$ we define a $k$-simplex $sigma$ to be $sigma={sum_{i=1}^kt_ia_i|t_0+...+t_k=1, t_ige 0}$. A simplicial complex $K$ is then a finite collection of simplices in some $mathbb{R}^n$ such that a) $K$ contains all faces of a simplex $sigmain K$ and b) the intersection of two simplices is a face of each of them.

I have two questions:

1) Why do we need the condition a)? What confuses me is that, in particular taking geometric realization, one can not distinguish e.g. between the standard simplex and the union of all its faces. Is there a historical reason for this formalism?

2) What if we defined $sigma={sum_{i=1}^kt_ia_i|t_0+...t_k=1, t_i > 0}$, a somewhat open analogue of the above definition, and take simplicial complexes of open simplices? This seems to be more natural to me considering the geometric realization (we don't take point-sets twice), and the notion of the interior and closure of a simplex is much more natural. However, this approach seems to be quite rare in the literature. Is there a good reason to consider closed simplices and do both theories coincide?

Good question! In terms of the actual set/shape in $mathbb{R}^n$, there is no difference, as you say. And you could do the "open simplex" thing, including all the boundaries as lower-dimensional open simplices, to get rid of repeats, without changing the union of all of them. But let me argue for the textbook definition.

I don't know much about this stuff, but one clear reason for (a) and (b) from my point of view is the correspondence between facts about the geometric realization of the simplicial complex and facts about the combinatorial object $mathcal{K}$. There are lots of topological properties of the geometric realization $|mathcal K|$ which can be computed or checked simply by knowing the set $mathcal K$ along with the subset/intersection structure, forgetting that each element once had geometry. A notable example: simplicial homology seems very simple compared to other homology theories (e.g., singular homology), and it agrees with the others. That's pretty amazing, and it also really helps with building intuition in the wobbly world of topology when you can take some reasonable geometric object you want to learn about, reduce it to a seemingly much simpler combinatorial object, and do computations there (these kinds of stories are some of my favorite parts of math). These correspondences depend on these two axioms (a) and (b) on the combinatorial side of the story. They are the data you need to keep to be able to study the topology (homology, in particular) of $|mathcal K|$ combinatorially.

The point of a simplicial complex is to create a combinatorial structure that one can analyze in place of a geometric space. If you were to break $K$ up into $K^0,K^1,dots,K^N$ with each $K^k$ containing all the $k$-simplicies from $K$, then knowing just the map $partial:K^{k+1}to mathcal{P}(K^{k})$ that carries a simplex to its boundary faces lets one construct a geometric realization. You don't need to know anything about the simplices themselves -- just how they are "glued" together. (Note: $K$ and the $K^k$'s contain the simplicies as elements and not as subsets.)

So for (a), one part is that without this condition you cannot reliably construct a geometric realization from only the data given by $partial$, and another is that even if it were reliable for a particular $K$, you can always replace it with the unique maximal $K$ that contains all the boundary faces ("since you can you must" is not uncommon in mathematical definitions). In practice, you can describe a simplicial complex with less information than what is required by the full definition.

For your second question, it is natural to used closed simplicies in the way quotient spaces are used to construct geometric realizations. The way that the boundaries coincide gives the particular identification. However, you are correct that for this particular definition of simplicial complex (as an actual subspace of $mathbb{R}^N$) you can use open simplices --- but you ought to include something like that the closure of the $(k+1)$-skeleton lies within the $k$-skeleton, like the definition of a CW decomposition of a space.

I'm going to give an answer to both of these questions using some knowledge of algebraic topology, because I feel that it where the real power of simplicial complexes and their various properties come into play.

For the first question I would ask the following counter-question, why would you want to distinguish between the standard simplex and the union of all its faces? Take this analogy, consider any set $X$, then $X$ is a union of all the points in $X$, would we want to distinguish between $bigcup_{p in X} {p}$ and $X$?

In fact when learning about simplices I actually found the fact that a simplex was the union of all its faces to be really useful and intuitive, this is particularly nice because if we are given say a $k$-simplicial complex $B$ we can look at it's $n$-skeleton which is intuitively the collection of simplices of dimensions less than or equal to $n$ sitting inside $B$.

The $k$-skeleton of the $n$-dimensional simplical complex $K$ turns out to be just the simplicial complex $K$ itself which is nice but also useful $(*)$. Using simplicial homology the skeletons of a simplicial complex actually tells us a ton of information about the topological space (the geometric realization of the simplicial complex) we are working with.

So let me give a quick example to show why $(*)$ is useful. Let's say we have a topological space $X$, with triangulation a simplicial complex $K$ of dimension $n$ and another topological space $Y$ with triangulation a simplicial complex $L$ of dimension $m$. Let's say that $m < n$, then $H_{m+1}(Y) = 0$ but it may be the case that $H_{m+1}(X) neq 0$ and if that's the case then since the homology groups of a topological space are an invariant of the topological space, because the homology groups of these two topological spaces aren't isomorphic the two topological spaces $X$ and $Y$ aren't homeomorphic.

Now had we made the distinction you stated this wouldn't necessarily be the case, the reason being that the $m$-skeleton of $L$ wouldn't be the same as $L$ and so the geometric realization of the $m$-skeleton (which is $|L|$ in reality) wouldn't necessarily be homeomorphic to $Y$, and then we wouldn't necessarily get $H_{m+1}(L) cong H_{m+1}(|L|) = H_{m+1}(Y)$ and so we wouldn't be able to conclude that $H_{m+1}(Y) = 0$ which basically throws a very nice result of simplicial homology away.

For the second question off the bat I think the first thing we would lose is compactness of simplices. Using the conventional defintion, all compact connected surfaces, like the torus $mathbb{T}$, real projective plane $mathbb{RP}^2$, the $2$-dimensional sphere $mathbb{S}^2$, are simplicial complexes meaning loosely that they can be built from simplices. However using your proposed definition it may not be the case that these very important topological spaces end up being simplicial complexes.

This is a big deal because the machinery of simplicial homology was essentially developed to calculate the homology of simplicial complexes and actually help us tell which of these spaces are different from each other (i.e. to tell which of these spaces are not homeomorphic to each other).

1)
Conditions a) and b) are kind of the definition of the simplicial complex. If you drop them, you have again a collection of simplices. This may be useful, but you do not need to define a simplicial complex when you need a set of simplices.

2)
That's related to the question, why we define a simplicial complex and do not just use a set of (possibly open) simplices.

An example for the use of simplicial complexes is the Discrete Exterior Calculus. The DEC uses a simplicial complex together with differential forms to define discrete differential operators.
In the DEC, the differential is defined using Stoke's theorem: $int_Omega text{d}omega = int_{partialOmega} omega$.

And when $Omega$ is a subset of the $K_2$ sceleton (i.e. the set of 2-simplices) of your complex, then $partial Omega$ is a subset of the $K_1$ sceleton. By the definition of the complex using a) and b), the discrete boundary operator $partial$ for a set of triangles and edges is just the adjacency matrix between triangles and edges in the complex.

Dropping a), you would not have the boundaries in your complex at all and dropping b) you would have an open set for each triangle, which by definition does not include the boundary.

To summarize: A simplicial complex is a handy definition for different applications.
You may want to use another definition for your application, but then using the name simplicial complex may be misleading.