Simplexes, Star Convex
Star Convex
A set S is star convex, or star shaped, if there is a point x in S
such that S contains the segment joining x and y whenever S contains y.
The term "star" comes from the shape of a star,
which is not convex,
but is star convex, by placing x in the center.
If S is convex, then S is star convex with any x acting as the center.
If x belongs to a simplex, or several simplexes,
then the union of those simplexes is star convex,
with x acting as the center.
Closed Star
If x is a point in a complex K, the closed star at x is the union of the simplexes containing x.
This is a subcomplex of K, and is closed in K.
It is also star convex, as described above.
Open Star
Let x be a point in a complex K, and restrict attention to its closed star U.
Remove any lower dimensional simplexes (i.e. faces edges etc) of U that do not contain x,
and call the result U′.
Since U′ is open in every simplex,
it is open in U, and in K.
Every point that was removed from U is a limit point of U′, hence U is the closure of U′.
Finally, U′ is star convex.
Hence it is called the open star of x.
If V is a subcomplex of U,
the closed star of x (within V) is the closed star of x intersect U.
The same is true for open stars.
Restrict attention to one simplex,
and the same faces and edges (i.e. those not containing x) are removed, whether that simplex is part of U or V.
The interior of the closed star is not always the open star.
Place x at the corner of a triangle, which is the entire complex K.
The closed star is K, and since K is an open set, the interior is K.
Yet the open star is K minus the far edge.
Link of x
The closed star of x minus the open star of x is the link of x.
This is a closed set minus an open set, hence it is closed.
If x is an isolated vertex,
the closed star equals the open star equals x,
and the link is trivial.
If x is part of a nontrivial simplex,
there is always a vertex apart from x, and the link is nontrivial.
Locally Path Connected
For any x in K, let G be the open star of x.
By passing through x, any two points in G are connected.
Therefore K is locally path connected.
This means components and path components are synonymous.
There are topological spaces,
such as the topologist's sine curve,
that are not locally path connected,
and cannot be realized by a simplicial complex.
In other words, the curve is not homeomorphic to a simplicial complex.
If x belongs to an open set in a complex K,
restrict the open set to the open star about x.
Further restrict this to an open ball about x.
One can contract the entire open star linearly onto x.
Any loop within the open ball, within the open star, contracts down to x as well.
It remains within the open star, and within K, at all times.
Therefore K is locally simply connected.
This means a space that is not locally simply connected,
such as the descending earring,
cannot be realized by a simplicial complex.
Since K is locally simply connected, K has a
universal cover U.
Each simplex of K is, by definition, homeomorphic to a simplex floating in free space.
In other words, different faces of one simplex are not sewn together,
as is done with the square to make a torus.
Therefore each closed simplex in K is contractable to a point and simply connected.
Lift any simplex up to U and find a homeomorphic copy of itself.
Furthermore, adjacent simplexes, sharing a common face, lift to adjacent simplexes in U.
Therefore U is simplicial,
and the projection is a simplicial map from U onto K.
The abstract map carries the fiber of each vertex onto that vertex.
This holds for any covering space, not just the universal cover.
The open stars of K, centered at the vertices, form an open cover for K.
If y is not a vertex it is part of a minimal simplex, with vertex x,
and is pulled in by the open star at x.
Each star pulls in its own vertex, and no other.
Another vertex y is not in the closed star, or it is in a face opposite to x, which has been removed.
Therefore a complex with infinitely many vertices is not compact.
Since each simplex has finitely many vertices, a complex K has infinitely many vertices iff it has infinitely many simplexes,
i.e. an infinite complex.
Conversely, a finite complex embeds in a finite slice of Ej,
which is homeomorphic to real space.
This embedding is closed and bounded, hence K is compact.
In summary, K is compact iff K is finite.