571: For Simplicial Complex, Simplex Interior of Maximal Simplex Does Not Intersect Any Other Simplex

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description/proof of that for simplicial complex, simplex interior of maximal simplex does not intersect any other 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.
• The reader knows a definition of simplex interior of affine simplex.
• The reader knows a definition of maximal simplex in 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, the simplex interior of any maximal simplex does not intersect any other 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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$S_k$: $\in \{\text{ the maximal simplexes in }C\}$
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Statements:
$\mathrm{\forall }{S}_j\in C\setminus \{S_k\}(S_k^{\circ }\cap S_j=\mathrm{\varnothing })$.
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2: Natural Language Description

For any real vectors space, $V$, any simplicial complex, $C$, on $V$, and any maximal simplex, $S_k\in C$, the simplex interior of $S_k$, $S_k^{\circ }$, does not intersect any other simplex, $S_j\in C\setminus \{S_k\}$, which is $S_k^{\circ }\cap S_j=\mathrm{\varnothing }$.

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3: Proof

Let us suppose that $S_k^{\circ }\cap S_j\ne \mathrm{\varnothing }$.

There would be a point, $p^{\prime }\in S_k^{\circ }\cap S_j$. $p^{\prime }\in S_k\cap S_j$. $S_k\cap S_j$ would be a face of $S_k$ that contains $p^{\prime }$. But any proper face of $S_k$ would not contain $p^{\prime }$, because $p^{\prime }\in S_k^{\circ }$. So, $S_k\cap S_j=S_k$, the non-proper face. While $S_k\cap S_j=S_k$ was also a face of $S_j$, as $S_k$ was maximal, $S_k$ would be the non-proper face of $S_j$, $S_j$. So, $S_k=S_j$, a contradiction against $S_j\ne S_k$.

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References

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