description/proof of that element of simplicial complex on finite-dimensional real vectors space is closed and compact on underlying space of complex
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 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.
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.
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:
\(V\): \(\in \{\text{ the } d \text{ -dimensional real vectors spaces }\}\)
\(C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(\vert C \vert\): \(= \text{ the underlying space of } C\)
//
Statements:
\(\forall S_\alpha \in C (S_\alpha \in \{\text{ the closed subsets of } \vert C \vert\} \cap \{\text{ the compact subsets of } \vert C \vert\})\)
//
2: Natural Language Description
For any \(d\)-dimensional vectors space, \(V\), any simplicial complex, \(C\), on \(V\), and the underling space, \(\vert C \vert\), of \(C\), each \(S_\alpha \in C\) is a closed and compact subset of \(\vert C \vert\).
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 \(S_\alpha = S_\alpha \cap \vert C \vert\), \(S_\alpha\) is a closed subset of \(\vert C \vert\).
\(S_\alpha\) is compact on \(\vert C \vert\), by the proposition that for any topological space, any subspace subset that is compact on the base space is compact on the subspace.
