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
Topics
About: vectors space
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
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.
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.
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
\(C\): \(\in \{\text{ the finite simplicial complexes on } V\}\)
\(\vert C \vert\): \(= \text{ the underlying space of } C\)
\(S_\alpha\): \(\in \{\text{ the maximal simplexes in } C\}\)
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Statements:
\(S_\alpha^\circ \in \{\text{ the open subsets of } \vert C \vert\}\).
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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 \(C\), \(\vert C \vert\).
3: Proof
For any \(p \in S_\beta^\circ\), \(p \notin S_\alpha\) for any \(S_\alpha \in C\) such that \(S_\alpha \neq S_\beta\), because \(S_\beta^\circ \cap S_\alpha = \emptyset\), 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 \(\vert C \vert\), 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, \(U_{p, \alpha} \subseteq \vert C \vert\), of \(p\) on \(\vert C \vert\) such that \(U_{p, \alpha} \cap S_\alpha = \emptyset\).
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, \(U'_{p, \beta} \subseteq V\), of \(p\) on \(V\) such that \(U'_{p, \beta} \cap S_\beta \subseteq S_\beta^\circ\), while \(U_{p, \beta} := U'_{p, \beta} \cap \vert C \vert\) is open on \(\vert C \vert\) and \(U_{p, \beta} \cap S_\beta = U'_{p, \beta} \cap S_\beta \subseteq S_\beta^\circ\).
Let us take \(U_p := \cap_{S_\alpha \in C} U_{p, \alpha} \subseteq \vert C \vert\), which is an open neighborhood of \(p\) on \(\vert C \vert\), because \(C\) has only some finite elements. \(U_p \cap S_\alpha = \emptyset\) for \(\alpha \neq \beta\), because \(U_p \subseteq U_{p, \alpha}\) and \(U_{p, \alpha} \cap S_\alpha = \emptyset\). \(U_p = U_p \cap \vert C \vert = U_p \cap \cup_{S_\alpha \in C} S_\alpha = U_p \cap (S_\beta \cup \cup_{S_\alpha \in C \setminus \{S_\beta\}} S_\alpha) = (U_p \cap S_\beta) \cup (U_p \cap \cup_{S_\alpha \in C \setminus \{S_\beta\}} S_\alpha) = (U_p \cap S_\beta) \cup (\cup_{S_\alpha \in C \setminus \{S_\beta\}} (U_p \cap S_\alpha)) = U_p \cap S_\beta \subseteq U_{p, \beta} \cap S_\beta \subseteq S_\beta^\circ\).
By the local criterion for openness, \(S_\beta^\circ\) is open on \(\vert C \vert\).
4: Note
When \(C\) is not finite, \(S_\beta^\circ\) may not be open on \(\vert C \vert\). As a counterexample, let \(V = \mathbb{R}^2\), the Euclidean vectors space, and \(C\) consist of affine 1-simplexes (with their faces); \(S_0 = [(0, 0), (1, 0)]\); \(\{S_j = [(0, 0), (1, 1 / j)] \vert j \in \mathbb{N} \setminus \{0\}\}\); then, for \(p = (1 / 2, 0)\), whatever open ball, \(B_{p, \epsilon} \subseteq \mathbb{R}^2\), 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.
