description/proof of that for simplicial complex, vertex of simplex that is on another simplex is vertex of latter 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.
Target Context
- The reader will have a description and a proof of the proposition that for any simplicial complex, each vertex of each simplex that is on any another simplex is a vertex of the latter 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:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(S_k\): \(\in C\)
\(p\): \(\in \{\text{ the vertexes of } S_k\}\)
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Statements:
\(\forall S_j \in C \text{ such that } p \in S_j (p \in \{\text{ the vertexes of } S_j\})\).
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2: Natural Language Description
For any real vectors space, \(V\), any simplicial complex, \(C\), on \(V\), any simplex, \(S_k \in C\), and any vertex, \(p \in S_k\), for each simplex, \(S_j \in C\), such that \(p \in S_j\), \(p\) is a vertex of \(S_j\).
3: Proof
\(\{p\}\) is a simplex contained in \(C\), because it is a face of \(S_k\), by the definition of simplicial complex. \(\{p\} \cap S_j = \{p\}\), which is a face of \(S_j\), by the definition of simplicial complex, which means that \(p\) is a vertex (a 0-face) of \(S_j\).
