Definition of star in a simplicial complex

Given a simplicial complex K and a collection of simplices S in K, the star of S is defined as the set of all simplices that have a face in S. Now consider the following picture (from wikipedia):

In the picture, S consists of the yellow point and its star is shown in green on the right.

Now it appears that this is somewhat of a convention, but the edges which are opposite of S in the green triangles are not part of the star of S. I find this confusing because I thought that they were parts of 2-simplices, the triangles which are shaded. The edges, when considered as 1-simplices in their own right, certainly do not contain S as a face, but the 2-simplices which contain them do contain S as a face.

So, a question. In diagrams like this, why do we exclude edges like this from the star? Aren't each of them a part of a 2-simplex which contains S?

Solution 1:

One needs to distinguish between an abstract simplicial complex and its geometric realization. We are dealing with the former here, so that we are thinking of a simplex as a collection of vertices. The correspondence to the geometric realization is probably best visualized by taking the interiors of simplices.

Solution 2:

I think your confusion might be in expecting the star of S to be closed in the sense: for every simplex in the star, the faces of that simplex should also be there.

The star is not defined in that way, and in general the star by itself is not a simplicial complex.

There is also no convention involved: you add the simplex if it has a face in S. As you point out, those edges do not contain S as a face, so you do not add them.