929: Union of Simplicial Complexes Is Not Necessarily Simplicial Complex

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description/proof of that union of simplicial complexes is not necessarily simplicial complex

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of simplicial complex.

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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.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V_1$: $\in \{\text{ the real vectors spaces }\}$
$V_2$: $\in \{\text{ the real vectors spaces }\}$
$C_1$: $\in \{\text{ the simplicial complexes on }{V}_1\}$
$C_2$: $\in \{\text{ the simplicial complexes on }{V}_2\}$
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Statements:
$V_1\in \{\text{ the subspaces of }{V}_2\}\vee V_2\in \{\text{ the subspaces of }{V}_1\}$
$⟹$
Not necessarily $C_1\cup C_2\in \{\text{ the simplicial complexes on }{V}_1\cup V_2\}$
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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 $V_1\cup V_2$.

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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 $S_1$.

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References

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