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
Topics
About: vectors space
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
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 \(C^\infty\) 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.
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.
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.
Main Body
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
\(C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(\vert C \vert\): \(= \text{ the underlying space of } C\)
\(U\): \(\subseteq \vert C \vert\), \(\in \{\text{ the open subsets of } \vert C \vert\}\)
\(st (p)\): \(= \text{ the star of } p\), where \(p \in Vert C\)
//
Statements:
\(U \cap st (p) \neq \emptyset\)
\(\implies\)
\(\exists S_k \in \{\text{ the maximal simplexes in } C\} (p \in Vert S_k \land S_k^\circ \cap U \neq \emptyset)\)
//
2: Natural Language Description
For any \(d\)-dimensional real vectors space, \(V\), any simplicial complex, \(C\), on \(V\), the underlying space, \(\vert C \vert\), any open subset, \(U \subset \vert C \vert\), and the star of any vertex, \(p \in Vert C\), \(st (p)\), if \(U \cap st (0) \neq \emptyset\), there is a maximal simplex, \(S_k \in C\), such that \(p \in Vert S_k\) and \(S_k^\circ \cap U \neq \emptyset\).
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 \(p' \in U \cap st (p)\), \(p'\) 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, \(p' \in S_k = [p_0 = p, p_1 ..., p_n]\), and \(p' = \sum_{j \in \{0, ..., n\}} t^j p_j\).
If \(S_l\) is \(S_k\) itself, \(p' \in S_k^\circ \cap U \neq \emptyset\).
Let us suppose that \(S_l\) is a proper face of \(S_k\) hereafter.
\(p'\)'s being on a proper face of \(S_k\) means that some of \(t^j\) s are \(0\) but \(0 \lt t^0\), because the face is \([p, p'_1 ..., p'_m]\) where \(\{p, p'_1, ..., p'_m\} \subseteq \{p, p_1, ..., p_n\}\) (the missing vertexes correspond to the \(0\) \(t^j\) s), and if \(t^0 = 0\), \(p'\) would not be on the simplex interior of \([p, p'_1 ..., p'_m]\). That means that \(\sum_{j \in \{1, ..., m\}} t^j \lt 1\), 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, \((V, \phi)\), \(\phi: s^1 (p_1 - p) + ... + s^n (p_n - p) + s^{n + 1} b_{n + 1} + ... + s^{d} b_{d} \mapsto (s^1, ..., s^{d})\).
As \(p' = t^0 p + \sum_{j \in \{1, ..., n\}} t^j p_j = \sum_{j \in \{1, ..., n\}} t^j (p_j - p) + t^0 p + \sum_{j \in \{1, ..., n\}} t^j p = \sum_{j \in \{1, ..., n\}} t^j (p_j - p) + \sum_{j \in \{0, ..., n\}} t^j p = \sum_{j \in \{1, ..., n\}} t^j (p_j - p) + p\), \(\phi (p') = (t^1 + p^1, ..., t^n + p^n, p^{n + 1}, ..., p^{d})\), where \(\phi (p) = (p^1, ..., p^{d})\).
As \(U\) is open on \(\vert C \vert\), there is an open ball, \(B_{p', \epsilon} \subseteq \vert C \vert\), around \(p'\) such that \(B_{p', \epsilon} \subseteq U\). For any point, \(p'' \in B_{p', \epsilon}\), \(\phi (p'') = (t^1 + \delta^1 + p^1, ..., t^n + \delta^n + p^n, p^{n + 1} + \delta^{n + 1}, ..., p^{d} + \delta^{d})\).
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 \(t^1 + \delta^1 + ... + t^n + \delta^n \lt 1\) and \(\sqrt{\sum_{j \in \{1, ..., d\}} \delta_j^2} \lt \epsilon\). 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 \(0 \lt t^j + \delta^j\). That means that \(p'' \in S_k^\circ\).
So, \(\emptyset \neq (B_{p', \epsilon} \cap S_k^\circ) \subseteq (U \cap S_k^\circ)\).
