578: For Finite Simplicial Complex on Finite-Dimensional Real Vectors Space, Simplex Interior of Maximal Simplex Is Open on Underlying Space of Complex

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description/proof of that for finite simplicial complex on finite-dimensional real vectors space, simplex interior of maximal simplex is open on underlying space of 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: Proof
• 4: Note

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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.
• The reader admits the proposition that for any simplicial complex, the simplex interior of any maximal simplex does not intersect any other simplex.
• The reader admits 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.
• The reader admits the proposition that the simplex interior of any affine simplex is open on the affine simplex with the canonical topology.
• The reader admits the local criterion for openness.

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Target Context

• The reader will have a description and a proof of the proposition that for any finite simplicial complex on any finite-dimensional real vectors space, the simplex interior of each maximal simplex is open 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:
$V$: $\in \{\text{ the }d\text{ -dimensional real vectors spaces }\}$ with the canonical topology
$C$: $\in \{\text{ the finite simplicial complexes on }V\}$
$|C|$: $=\text{ the underlying space of }C$
$S_{\alpha }$: $\in \{\text{ the maximal simplexes in }C\}$
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Statements:
$S_{\alpha }^{\circ }\in \{\text{ the open subsets of }|C|\}$.
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2: Natural Language Description

For any $d$-dimensional real vectors space, $V$, with the canonical topology, any finite simplicial complex, $C$, on $V$, and any maximal simplex, $S_{\alpha }\in C$, the simplex interior of $S_{\alpha }$, $S_{\alpha }^{\circ }$, is open on the underlying space of $C$, $|C|$.

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3: Proof

For any $p\in S_{\beta }^{\circ }$, $p\notin S_{\alpha }$ for any $S_{\alpha }\in C$ such that $S_{\alpha }\ne S_{\beta }$, because $S_{\beta }^{\circ }\cap S_{\alpha }=\mathrm{\varnothing }$, by the proposition that for any simplicial complex, the simplex interior of any maximal simplex does not intersect any other simplex.

As $S_{\alpha }$ is closed on $|C|$, by 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, there is an open neighborhood, $U_{p,\alpha }\subseteq |C|$, of $p$ on $|C|$ such that $U_{p,\alpha }\cap S_{\alpha }=\mathrm{\varnothing }$.

As $S_{\beta }^{\circ }$ is open on $S_{\beta }$, by the proposition that the simplex interior of any affine simplex is open on the affine simplex with the canonical topology, there is an open neighborhood, $U_{p,\beta }^{\prime }\subseteq V$, of $p$ on $V$ such that $U_{p,\beta }^{\prime }\cap S_{\beta }\subseteq S_{\beta }^{\circ }$, while $U_{p,\beta }:=U_{p,\beta }^{\prime }\cap |C|$ is open on $|C|$ and $U_{p,\beta }\cap S_{\beta }=U_{p,\beta }^{\prime }\cap S_{\beta }\subseteq S_{\beta }^{\circ }$.

Let us take $U_p:={\cap }_{S_{\alpha }\in C}{U}_{p,\alpha }\subseteq |C|$, which is an open neighborhood of $p$ on $|C|$, because $C$ has only some finite elements. $U_p\cap S_{\alpha }=\mathrm{\varnothing }$ for $\alpha \ne \beta$, because $U_p\subseteq U_{p,\alpha }$ and $U_{p,\alpha }\cap S_{\alpha }=\mathrm{\varnothing }$. $U_p=U_p\cap |C|=U_p\cap {\cup }_{S_{\alpha }\in C}{S}_{\alpha }=U_p\cap (S_{\beta }\cup {\cup }_{S_{\alpha }\in C\setminus \{S_{\beta }\}}{S}_{\alpha })=(U_p\cap S_{\beta })\cup (U_p\cap {\cup }_{S_{\alpha }\in C\setminus \{S_{\beta }\}}{S}_{\alpha })=(U_p\cap S_{\beta })\cup ({\cup }_{S_{\alpha }\in C\setminus \{S_{\beta }\}}(U_p\cap S_{\alpha }))=U_p\cap S_{\beta }\subseteq U_{p,\beta }\cap S_{\beta }\subseteq S_{\beta }^{\circ }$.

By the local criterion for openness, $S_{\beta }^{\circ }$ is open on $|C|$.

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4: Note

When $C$ is not finite, $S_{\beta }^{\circ }$ may not be open on $|C|$. As a counterexample, let $V={\mathbb{R}}^2$, the Euclidean vectors space, and $C$ consist of affine 1-simplexes (with their faces); $S_0=[(0,0),(1,0)]$; $\{S_j=[(0,0),(1,1/j)]|j\in \mathbb{N}\setminus \{0\}\}$; then, for $p=(1/2,0)$, whatever open ball, $B_{p,ϵ}\subseteq {\mathbb{R}}^2$, intersects an $S_j$, because $S_j$ nears $S_0$ infinitely as $j$ increases.

While we have proved another proposition that for any finite simplicial complex on any finite-dimensional real vectors space, the simplex interior of each complex-dimensional element is open on the underlying space of the complex, this proposition states that each maximal, not necessarily complex-dimensional, simplex is open: each complex-dimensional simplex is obviously a maximal simplex, but a maximal simplex is not necessarily complex-dimensional.

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References

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