931: For Simplicial Complex and Its Subcomplexes, Union of Subcomplexes Is Simplicial Complex

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description/proof of that for simplicial complex and its subcomplexes, union of subcomplexes is 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: Note
• 4: 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 for any simplicial complex and its any subcomplexes, the union of the subcomplexes is 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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$C^{\prime }$: $\in \{\text{ the simplicial complexes on }V\}$
$C_1$: $\in \{\text{ the subcomplexes of }{C}^{\prime }\}$
$C_2$: $\in \{\text{ the subcomplexes of }{C}^{\prime }\}$
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Statements:
$C_1\cup C_2\in \{\text{ the simplicial complexes on }V\}$
//

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2: Natural Language Description

For any real vectors space, $V$, any simplicial complex, $C^{\prime }$, on $V$, and any subcomplexes of $C^{\prime }$, $C_1,C_2$, $C_1\cup C_2$ is a simplicial complex on $V$.

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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.

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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. $S_1,S_2\in C^{\prime }$. So, $S_1\cap S_2$ is a face of $S_1$ and a face of $S_2$.

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References

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