description/proof of that for simplicial complex, simplex interior of maximal simplex does not intersect any other simplex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of simplicial complex.
- The reader knows a definition of simplex interior of affine simplex.
- The reader knows a definition of maximal simplex in simplicial complex.
Target Context
- The reader will have a description and a proof of the proposition that for any simplicial complex, the simplex interior of any maximal simplex does not intersect any other simplex.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(S_k\): \(\in \{\text{ the maximal simplexes in } C\}\)
//
Statements:
\(\forall S_j \in C \setminus \{S_k\} (S_k^\circ \cap S_j = \emptyset)\).
//
2: Natural Language Description
For any real vectors space, \(V\), any simplicial complex, \(C\), on \(V\), and any maximal simplex, \(S_k \in C\), the simplex interior of \(S_k\), \(S_k^\circ\), does not intersect any other simplex, \(S_j \in C \setminus \{S_k\}\), which is \(S_k^\circ \cap S_j = \emptyset\).
3: Proof
Let us suppose that \(S_k^\circ \cap S_j \neq \emptyset\).
There would be a point, \(p' \in S_k^\circ \cap S_j\). \(p' \in S_k \cap S_j\). \(S_k \cap S_j\) would be a face of \(S_k\) that contains \(p'\). But any proper face of \(S_k\) would not contain \(p'\), because \(p' \in S_k^\circ\). So, \(S_k \cap S_j = S_k\), the non-proper face. While \(S_k \cap S_j = S_k\) was also a face of \(S_j\), as \(S_k\) was maximal, \(S_k\) would be the non-proper face of \(S_j\), \(S_j\). So, \(S_k = S_j\), a contradiction against \(S_j \neq S_k\).
