577: Element of Simplicial Complex on Finite-Dimensional Real Vectors Space Is Closed and Compact on Underlying Space of Complex

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description/proof of that element of simplicial complex on finite-dimensional real vectors space is closed and compact on underlying space of complex

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Topics

About: vectors space
About: topological 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 closed set.
• The reader knows a definition of compact subset of topological space.
• The reader knows a definition of subspace topology of subset of topological space.
• The reader admits the proposition that any affine simplex on any finite-dimensional real vectors space is closed and compact on the canonical topological superspace.
• The reader admits the proposition that for any topological space, any subspace subset that is compact on the base space is compact on the subspace.

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Target Context

• The reader will have a description and a proof of the proposition that each element of any simplicial complex on any finite-dimensional real vectors space is closed and compact on the underlying space of the 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:
$V$: $\in \{\text{ the }d\text{ -dimensional real vectors spaces }\}$
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$|C|$: $=\text{ the underlying space of }C$
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Statements:
$\mathrm{\forall }{S}_{\alpha }\in C(S_{\alpha }\in \{\text{ the closed subsets of }|C|\}\cap \{\text{ the compact subsets of }|C|\})$
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2: Natural Language Description

For any $d$-dimensional vectors space, $V$, any simplicial complex, $C$, on $V$, and the underling space, $|C|$, of $C$, each $S_{\alpha }\in C$ is a closed and compact subset of $|C|$.

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3: Proof

As $S_{\alpha }$ is an affine simplex on $V$, $S_{\alpha }$ is closed and compact on $V$, by the proposition that any affine simplex on any finite-dimensional real vectors space is closed and compact on the canonical topological superspace.

As $S_{\alpha }=S_{\alpha }\cap |C|$, $S_{\alpha }$ is a closed subset of $|C|$.

$S_{\alpha }$ is compact on $|C|$, by the proposition that for any topological space, any subspace subset that is compact on the base space is compact on the subspace.

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References

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