570: For Simplicial Complex, Vertex of Simplex That Is on Another Simplex Is Vertex of Latter Simplex

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description/proof of that for simplicial complex, vertex of simplex that is on another simplex is vertex of latter simplex

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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 for any simplicial complex, each vertex of each simplex that is on any another simplex is a vertex of the latter simplex.

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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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$S_k$: $\in C$
$p$: $\in \{\text{ the vertexes of }{S}_k\}$
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Statements:
$\mathrm{\forall }{S}_j\in C\text{ such that }p\in S_j(p\in \{\text{ the vertexes of }{S}_j\})$.
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2: Natural Language Description

For any real vectors space, $V$, any simplicial complex, $C$, on $V$, any simplex, $S_k\in C$, and any vertex, $p\in S_k$, for each simplex, $S_j\in C$, such that $p\in S_j$, $p$ is a vertex of $S_j$.

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3: Proof

$\{p\}$ is a simplex contained in $C$, because it is a face of $S_k$, by the definition of simplicial complex. $\{p\}\cap S_j=\{p\}$, which is a face of $S_j$, by the definition of simplicial complex, which means that $p$ is a vertex (a 0-face) of $S_j$.

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References

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