584: For Simplicial Complex on Finite-Dimensional Real Vectors Space, Open Subset of Underlying Space That Intersects Star Intersects Simplex Interior of Maximal Simplex Involved in Star

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description/proof of that for simplicial complex on finite-dimensional real vectors space, open subset of underlying space that intersects star intersects simplex interior of maximal simplex involved in star

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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 simplex interior of affine simplex.
• The reader knows a definition of maximal simplex in simplicial complex.
• The reader knows a definition of star of vertex in simplicial complex.
• The reader knows a definition of topological subspace.
• The reader knows a definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space.
• The reader admits the proposition that for any simplicial complex on any finite-dimensional real vectors space, each simplex in the complex is the faces of the elements of a subset of the maximal simplexes set.

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Target Context

• The reader will have a description and a proof of the proposition that for any simplicial complex on any finite-dimensional real vectors space, any open subset of the underlying space that intersects any star intersects the simplex interior of a maximal simplex involved in the star.

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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
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$|C|$: $=\text{ the underlying space of }C$
$U$: $\subseteq |C|$, $\in \{\text{ the open subsets of }|C|\}$
$st(p)$: $=\text{ the star of }p$, where $p\in VertC$
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Statements:
$U\cap st(p)\ne \mathrm{\varnothing }$
$⟹$
$\mathrm{\exists }{S}_k\in \{\text{ the maximal simplexes in }C\}(p\in Vert{S}_k\wedge S_k^{\circ }\cap U\ne \mathrm{\varnothing })$
//

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2: Natural Language Description

For any $d$-dimensional real vectors space, $V$, any simplicial complex, $C$, on $V$, the underlying space, $|C|$, any open subset, $U\subset |C|$, and the star of any vertex, $p\in VertC$, $st(p)$, if $U\cap st(0)\ne \mathrm{\varnothing }$, there is a maximal simplex, $S_k\in C$, such that $p\in Vert{S}_k$ and $S_k^{\circ }\cap U\ne \mathrm{\varnothing }$.

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3: Proof

$st(p)$ involves all the maximal simplexes that have $p$ as a vertex (there are some positive number of such maximal simplexes), and any other simplex involved in $st(p)$ is a proper face of at least 1 of the maximal simplexes, by the proposition that for any simplicial complex on any finite-dimensional real vectors space, each simplex in the complex is the faces of the elements of a subset of the maximal simplexes set.

For any $p^{\prime }\in U\cap st(p)$, $p^{\prime }$ is on the simplex interior of a simplex, $S_l\in C$, involved in $st(p)$. $S_l$ is a (not necessarily proper) face of a maximal simplex, $S_k\in C$, involved in $st(p)$.

So, $p^{\prime }\in S_k=[p_0=p,p_1...,p_n]$, and $p^{\prime }=\sum _{j\in \{0,...,n\}}{t}^j{p}_j$.

If $S_l$ is $S_k$ itself, $p^{\prime }\in S_k^{\circ }\cap U\ne \mathrm{\varnothing }$.

Let us suppose that $S_l$ is a proper face of $S_k$ hereafter.

$p^{\prime }$'s being on a proper face of $S_k$ means that some of $t^j$ s are $0$ but $0<t^0$, because the face is $[p,p_1^{\prime }...,p_m^{\prime }]$ where $\{p,p_1^{\prime },...,p_m^{\prime }\}\subseteq \{p,p_1,...,p_n\}$ (the missing vertexes correspond to the $0$ $t^j$ s), and if $t^0=0$, $p^{\prime }$ would not be on the simplex interior of $[p,p_1^{\prime }...,p_m^{\prime }]$. That means that $\sum _{j\in \{1,...,m\}}{t}^j<1$, because $\sum _{j\in \{0,...,m\}}{t}^j=1$.

Let us take a basis of $V$ as $\{p_1-p,...,p_n-p,b_{n+1},...,b_d\}$ and the canonical chart, $(V,\varphi )$, $\varphi :s^1(p_1-p)+...+s^n(p_n-p)+s^{n+1}{b}_{n+1}+...+s^d{b}_d\mapsto (s^1,...,s^d)$.

As $p^{\prime }=t^{0}p+\sum _{j\in \{1,...,n\}}{t}^j{p}_j=\sum _{j\in \{1,...,n\}}{t}^j(p_j-p)+t^{0}p+\sum _{j\in \{1,...,n\}}{t}^{j}p=\sum _{j\in \{1,...,n\}}{t}^j(p_j-p)+\sum _{j\in \{0,...,n\}}{t}^{j}p=\sum _{j\in \{1,...,n\}}{t}^j(p_j-p)+p$, $\varphi (p^{\prime })=(t^1+p^1,...,t^n+p^n,p^{n+1},...,p^d)$, where $\varphi (p)=(p^1,...,p^d)$.

As $U$ is open on $|C|$, there is an open ball, $B_{p^{\prime },ϵ}\subseteq |C|$, around $p^{\prime }$ such that $B_{p^{\prime },ϵ}\subseteq U$. For any point, $p^{″}\in B_{p^{\prime },ϵ}$, $\varphi (p^{″})=(t^1+{\delta }^1+p^1,...,t^n+{\delta }^n+p^n,p^{n+1}+{\delta }^{n+1},...,p^d+{\delta }^d)$.

While some of $t^j$ s are $0$, the corresponding ${\delta }^j$ s can be taken to be small enough positive and the other ${\delta }^j$ s can be taken to be $0$ such that $t^1+{\delta }^1+...+t^n+{\delta }^n<1$ and $\sqrt{\sum _{j\in \{1,...,d\}}{\delta }_j^2}<ϵ$. Then, $p^{″}=\sum _{j\in \{1,...,n\}}(t^j+{\delta }^j+p^j)(p_j-p)+\sum _{j\in \{n+1,...,d\}}{p}^j{b}_j=\sum _{j\in \{1,...,n\}}(t^j+{\delta }^j)(p_j-p)+\sum _{j\in \{1,...,n\}}{p}^j(p_j-p)+\sum _{j\in \{n+1,...,d\}}{p}^j{b}_j=\sum _{j\in \{1,...,n\}}(t^j+{\delta }^j)(p_j-p)+p=\sum _{j\in \{1,...,n\}}(t^j+{\delta }^j)(p_j-p)+\sum _{j\in \{0,...,n\}}(t^j+{\delta }^j)p$, where ${\delta }^0$ is defined such that $\sum _{j\in \{0,...,n\}}(t^j+{\delta }^j)=1$, $=(t^0+{\delta }^0)p+\sum _{j\in \{1,...,n\}}(t^j+{\delta }^j)p_j$. $\sum _{j\in \{0,...,n\}}(t^j+{\delta }^j)=1$ and $0<t^j+{\delta }^j$. That means that $p^{″}\in S_k^{\circ }$.

So, $\mathrm{\varnothing }\ne (B_{p^{\prime },ϵ}\cap S_k^{\circ })\subseteq (U\cap S_k^{\circ })$.

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References

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