555: Simplicial Complex

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definition of 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

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Starting Context

• The reader knows a definition of affine simplex.

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Target Context

• The reader will have a definition of 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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\ast C$: $=\{S_{\alpha }|\alpha \in A,S_{\alpha }\in \{\text{ the affine simplexes on }V\}\}$, where $A$ is any possibly uncountable index set
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Conditions:
1) $\mathrm{\forall }{S}_{\alpha }\in C(\mathrm{\forall }{S}_j\in \{\text{ the faces of }{S}_{\alpha }\}(S_j\in C))$
$\wedge$
2) $\mathrm{\forall }{S}_{\alpha },S_{\beta }\in C\text{ such that }{S}_{\alpha }\cap S_{\beta }\ne \mathrm{\varnothing }(S_{\alpha }\cap S_{\beta }\in \{\text{ the faces of }{S}_{\alpha }\}\cap \{\text{ the faces of }{S}_{\beta }\})$
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When $V$ is finite-dimensional (although the definition does not suppose so in order not to impose any unnecessary assumption, usually, $V$ is supposed to be finite-dimensional), $|C|:={\cup }_{S_{\alpha }\in C}{S}_{\alpha }\subseteq V$, with the subspace topology of the canonical topology of $V$, is called underlying space of $C$.

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2: Natural Language Description

For any real vectors space, $V$, any set of some affine simplexes on $V$, $C=\{S_{\alpha }|\alpha \in A,S_{\alpha }\in \{\text{ the affine simplexes on }V\}\}$, where $A$ is any possibly uncountable index set, such that 1) $\mathrm{\forall }{S}_{\alpha }\in C(\mathrm{\forall }{S}_j\in \{\text{ the faces of }{S}_{\alpha }\}(S_j\in C))$ and 2) $\mathrm{\forall }{S}_{\alpha },S_{\beta }\in C\text{ such that }{S}_{\alpha }\cap S_{\beta }\ne \mathrm{\varnothing }(S_{\alpha }\cap S_{\beta }\in \{\text{ the faces of }{S}_{\alpha }\}\cap \{\text{ the faces of }{S}_{\beta }\})$

When $V$ is finite-dimensional (although the definition does not suppose so in order not to impose any unnecessary assumption, usually, $V$ is supposed to be finite-dimensional), $|C|:={\cup }_{S_{\alpha }\in C}{S}_{\alpha }\subseteq V$, with the subspace topology of the canonical topology of $V$, is called underlying space of $C$.

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3: Note

When $A$ is finite, $C$ is called finite simplicial complex.

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References

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