definition of simplicial 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: Note
Starting Context
- The reader knows a definition of affine simplex.
Target Context
- The reader will have a definition of simplicial 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\(*C\): \(= \{S_\alpha \vert \alpha \in A, S_\alpha \in \{\text{ the affine simplexes on } V\}\}\), where \(A\) is any possibly uncountable index set
//
Conditions:
1) \(\forall S_\alpha \in C (\forall S_j \in \{\text{ the faces of } S_\alpha\} (S_j \in C))\)
\(\land\)
2) \(\forall S_\alpha, S_\beta \in C \text{ such that } S_\alpha \cap S_\beta \neq \emptyset (S_\alpha \cap S_\beta \in \{\text{ the faces of } S_\alpha\} \cap \{\text{ the faces of } S_\beta\})\)
//
When \(V\) is finite-dimensional (although the definition does not suppose so in order not to impose any unnecessary assumption, usually, \(V\) is supposed to be finite-dimensional), \(\vert C \vert := \cup_{S_\alpha \in C} S_\alpha \subseteq V\), with the subspace topology of the canonical topology of \(V\), is called underlying space of \(C\).
2: Natural Language Description
For any real vectors space, \(V\), any set of some affine simplexes on \(V\), \(C = \{S_\alpha \vert \alpha \in A, S_\alpha \in \{\text{ the affine simplexes on } V\}\}\), where \(A\) is any possibly uncountable index set, such that 1) \(\forall S_\alpha \in C (\forall S_j \in \{\text{ the faces of } S_\alpha\} (S_j \in C))\) and 2) \(\forall S_\alpha, S_\beta \in C \text{ such that } S_\alpha \cap S_\beta \neq \emptyset (S_\alpha \cap S_\beta \in \{\text{ the faces of } S_\alpha\} \cap \{\text{ the faces of } S_\beta\})\)
When \(V\) is finite-dimensional (although the definition does not suppose so in order not to impose any unnecessary assumption, usually, \(V\) is supposed to be finite-dimensional), \(\vert C \vert := \cup_{S_\alpha \in C} S_\alpha \subseteq V\), with the subspace topology of the canonical topology of \(V\), is called underlying space of \(C\).
3: Note
When \(A\) is finite, \(C\) is called finite simplicial complex.
