description/proof of that when union of simplicial complexes is simplicial complex, underlying space of union is union of underlying spaces of constituents
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 when the union of some simplicial complexes is a simplicial complex, the underlying space of the union is the union of the underlying spaces of the constituent complexes.
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:
(
\(V_1 \in \{\text{ the subspaces of } V_2\} \lor V_2 \in \{\text{ the subspaces of } V_1\}\)
\(\land\)
\(C_1 \cap C_2 \in \{\text{ the simplicial complexes on } V_1 \cup V_2\}\)
)
\(\implies\)
\(\vert C_1 \cup C_2 \vert = \vert C_1 \vert \cup \vert C_2 \vert\)
//
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 \(V_2\), when \(C_1 \cup C_2\) is a simplicial complex on \(V_1 \cup V_2\), \(\vert C_1 \cup C_2 \vert = \vert C_1 \vert \cup \vert C_2 \vert\).
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. This is about when \(C_1 \cup C_2\) is so.
Compare with the proposition that the intersection of any 2 simplicial complexes is a simplicial complex, and the underlying space of the intersection is contained in but not necessarily equal to the intersection of the underlying spaces of the constituent simplicial complexes.
4: Proof
Whole Strategy: Step 1: see that \(\vert C_1 \cup C_2 \vert \subseteq \vert C_1 \vert \cup \vert C_2 \vert\); Step 2: see that \(\vert C_1 \vert \cup \vert C_2 \vert \subseteq \vert C_1 \cup C_2 \vert\).
Step 1:
Let \(p \in \vert C_1 \cup C_2 \vert\) be any. There is an \(S \in C_1 \cup C_2\) such that \(p \in S\). \(S \in C_1\) or \(S \in C_2\). \(p \in \vert C_1 \vert\) or \(p \in \vert C_2 \vert\), so, \(p \in \vert C_1 \vert \cup \vert C_2 \vert\).
Step 2:
Let \(p \in \vert C_1 \vert \cup \vert C_2 \vert\) be any. \(p \in \vert C_1 \vert\) or \(p \in \vert C_2 \vert\). There is an \(S_1 \in C_1\) such that \(p \in S_1\) or an \(S_2 \in C_2\) such that \(p \in S_2\). If the former holds, as \(S_1 \in C_1 \cup C_2\), \(p \in \vert C_1 \cup C_2 \vert\); otherwise, likewise, \(p \in \vert C_1 \cup C_2 \vert\), so, \(p \in \vert C_1 \cup C_2 \vert\) anyway.
