description/proof of that subset of underlying space of finite simplicial complex on finite-dimensional real vectors space is closed iff its intersection with each element of complex is closed
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 canonical topology for finite-dimensional real vectors space.
- The reader knows a definition of closed set.
- The reader knows a definition of subspace topology of subset of topological space.
- 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 for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset.
Target Context
- The reader will have a description and a proof of the proposition that any subset of the underlying space of any finite simplicial complex on any finite-dimensional real vectors space is closed iff its intersection with each element of the complex is closed.
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
\(C\): \(\in \{\text{ the finite simplicial complexes on } V\}\)
\(\vert C \vert\): \(= \text{ the underlying space of } C\)
\(S\): \(\subseteq \vert C \vert\)
//
Statements:
\(S \in \{\text{ the closed subsets of } \vert C \vert\}\)
\(\iff\)
\(\forall s \in C (s \cap S \in \{\text{ the closed subsets of } \vert C \vert\})\)
//
2: Natural Language Description
For any \(d\)-dimensional real vectors space, \(V\), with the canonical topology, any finite simplicial complex, \(C\), on \(V\), and the underling space, \(\vert C \vert\), of \(C\), any subset, \(S \subseteq \vert C \vert\), is closed if and only if for each \(s \in C\), \(s \cap S\) is closed on \(\vert C \vert\).
3: Proof
Let us suppose that \(S\) is closed on \(\vert C \vert\).
\(s\) 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. \(s \cap S\) is closed on \(\vert C \vert\) as the intersection of the closed subsets.
Let us suppose that \(s \cap S\) is closed on \(\vert C \vert\).
\(S = S \cap \vert C \vert = S \cap \cup_{s \in C} s = \cup_{s \in C} (s \cap S)\), by the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset. As \(s \cap S\) is closed on \(\vert C \vert\), the finite union, \(\cup_{s \in C} (s \cap S)\), is closed on \(\vert C \vert\).
