description/proof of that for simplicial complex and its subcomplexes, union of subcomplexes is 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: Note
- 4: 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 and its any subcomplexes, the union of the subcomplexes is 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:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(C'\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(C_1\): \(\in \{\text{ the subcomplexes of } C'\}\)
\(C_2\): \(\in \{\text{ the subcomplexes of } C'\}\)
//
Statements:
\(C_1 \cup C_2 \in \{\text{ the simplicial complexes on } V\}\)
//
2: Natural Language Description
For any real vectors space, \(V\), any simplicial complex, \(C'\), on \(V\), and any subcomplexes of \(C'\), \(C_1, C_2\), \(C_1 \cup C_2\) is a simplicial complex on \(V\).
3: Note
As has been shown in the proposition that the union of some simplicial complexes is not necessarily a simplicial complex, \(C_1 \cup C_2\) is not necessarily a simplicial complex, if \(C_1\) and \(C_2\) are not some subcomplexes of the same simplicial complex.
4: Proof
Whole Strategy: Step 1: see that \(C_1 \cup C_2\) satisfies the requirements to be a simplicial complex.
Step 1:
Let \(S \in C_1 \cup C_2\) be any. \(S \in C_1\) or \(S \in C_2\). Each face of \(S\) is contained in \(C_1\) or in \(C_2\), so, is contained in \(C_1 \cup C_2\).
Let \(S_1, S_2 \in C_1 \cup C_2\) be any. \(S_1, S_2 \in C'\). So, \(S_1 \cap S_2\) is a face of \(S_1\) and a face of \(S_2\).
