I am having trouble understanding the following from Robert Ghrist's book "Elementary Applied Topology" related to simplicial complexes, but first two definitions:
A family $ \Delta $ of non-empty finite subsets of a set S is an abstract simplicial complex if, for every set X in Δ, and every non-empty subset Y $\subset$ X, Y also belongs to $ \Delta $.
The standard n-simplex in $ \mathbb{R}^{n+1} $ is the following set: $ \Delta ^{n} = \{ (t_0,...,t_n) \in \mathbb{R}^{n+1} | \sum_{i=0}^{n} t_i = 1 \space \space \space \forall i: t_i \geq 0 \} $
Here is what I don't understand from Ghrist's book, from page 27:
How exactly are the faces of the standard simplex defined? which coordinates are nullified in the definition? I don't exactly understand coordinate subspace restrictions. I also don't understand the defintion of the skeleton given, how can they look at the disjoint union of standard simplices which seem to be the same, also I don't understand the exact definition of the equivalence relation ~ given below and why exactly we have nesting $ X^{(k)} \supset X^{(k-1)} $? I guess what really would help me understand this matter is a representative example of a skeleton and geometric realization of a specific abstract simplicial complex, would someone please help me, I thank all helpers.
Faces: For a $k$-simplex, a $k$ face is the whole simplex. The $k-1$ faces are those with one of the coordinates set to $0$. So for a 2-simplex, defined by $x+y+z = 1$, there are three 1-faces: $x+y = 1$ and $z = 0$; $x+z = 1$ and $y = 0$; and $y+z = 1$ and $x = 0$.
In general, the $k-p$ faces of a $k$-simplex are those in which exactly $p$ of the coordinates are set to 0. That means that there are $k \choose p$ of them.
For the further questions, just saying "I don't understand" isn't really enough for me to know WHAT you don't understand, so I cannot help there.