description/proof of that union of simplicial complexes is not necessarily simplicial complex
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 the union of some simplicial complexes is not necessarily a simplicial complex.
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:
\(V_1\): \(\in \{\text{ the real vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the real vectors spaces }\}\)
\(C_1\): \(\in \{\text{ the simplicial complexes on } V_1\}\)
\(C_2\): \(\in \{\text{ the simplicial complexes on } V_2\}\)
//
Statements:
\(V_1 \in \{\text{ the subspaces of } V_2\} \lor V_2 \in \{\text{ the subspaces of } V_1\}\)
\(\implies\)
Not necessarily \(C_1 \cup C_2 \in \{\text{ the simplicial complexes on } V_1 \cup V_2\}\)
//
2: Natural Language Description
For any real vectors spaces, \(V_1, V_2\), such that \(V_1\) is a vectors subspace of \(V_2\) or \(V_2\) is a vectors subspace of \(V_1\), and any simplicial complexes, \(C_1\), on \(V_1\) and \(C_2\), on \(V_2\), \(C_1 \cup C_2\) is not necessarily any simplicial complex on \(V_1 \cup V_2\).
3: Proof
Whole Strategy: Step 1: see a counterexample.
Step 1:
A counterexample suffices.
Let \(V_1 = V_2 = \mathbb{R}^2\) and \(C_1\) consist of a 2-simplex, \(S_1\), (and its faces) and \(C_2\) consist of a 2-simplex, \(S_2\), (and its faces) that shares a vertex and only a part of an edge (which contains the vertex) of \(S_1\). Then, while \(C_1 \cup C_2\) consists of \(S_1\) and \(S_2\) (and their faces), \(S_1 \cap S_2\) is the partial edge of \(S_1\), which is not any face of \(S_1\).
