description/proof of that for simplicial complex and its subcomplexes, underlying space of intersection of subcomplexes is intersection 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: Proof
Starting Context
- The reader knows a definition of simplicial complex.
- The reader admits 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.
Target Context
- The reader will have a description and a proof of the proposition that for any simplicial complex and its any subcomplexes, the underlying space of the intersection of the subcomplexes is the intersection of the underlying spaces of the constituent subcomplexes.
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:
\(C_1 \cap C_2 \in \{\text{ the simplicial complexes on } V\}\)
\(\land\)
\(\vert C_1 \cap C_2 \vert = \vert C_1 \vert \cap \vert C_2 \vert\)
//
2: Natural Language Description
For any real vectors space, \(V\), any simplicial complex, \(C'\), and any subcomplexes of \(C'\), \(C_1, C_2\), \(C_1 \cap C_2\) is a simplicial complex on \(V\), and \(\vert C_1 \cap C_2 \vert = \vert C_1 \vert \cap \vert C_2 \vert\).
3: Proof
Whole Strategy: Step 1: see that \(C_1 \cap C_2\) is a simplicial complex on \(V\) and \(\vert C_1 \cap C_2 \vert \subseteq \vert C_1 \vert \cap \vert C_2 \vert\); Step 2: see that \(\vert C_1 \vert \cap \vert C_2 \vert \subseteq \vert C_1 \cap C_2 \vert\).
Step 1:
This is a special case of 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, and so, \(C_1 \cap C_2\) is a simplicial complex on \(V\), and \(\vert C_1 \cap C_2 \vert \subseteq \vert C_1 \vert \cap \vert C_2 \vert\).
Step 2:
Let us see that \(\vert C_1 \vert \cap \vert C_2 \vert \subseteq \vert C_1 \cap C_2 \vert\).
Let \(p \in \vert C_1 \vert \cap \vert C_2 \vert\) be any. \(p \in \vert C_1 \vert\). There is an \(S_1 \in C_1\) such that \(p \in S_1\). Likewise, there is an \(S_2 \in C_2\) such that \(p \in S_2\). So, \(p \in S_1 \cap S_2\).
As \(S_1, S_2 \in C'\), \(S_1 \cap S_2\) is a face of \(S_1\). As \(S_1 \in C_1\), \(S_1 \cap S_2 \in C_1\). Likewise, \(S_1 \cap S_2 \in C_2\). So, \(S_1 \cap S_2 \in C_1 \cap C_2\). As \(p \in S_1 \cap S_2 \in C_1 \cap C_2\), \(p \in \vert C_1 \cap C_2 \vert\).
