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Module cell

koho::math

Module cell 

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Core traits and definitions for constructing a cell complex over arbitrary types.

This module provides the fundamental building blocks for topological spaces and CW complexes (cell complexes). It allows for the construction of spaces by attaching cells of various dimensions along their boundaries.

§Mathematical Background

§Topology

A topology on a set X is a collection of subsets (called open sets) that satisfies:

  • The empty set and X itself are open
  • Arbitrary unions of open sets are open
  • Finite intersections of open sets are open

§Cell Complex (CW Complex)

A CW complex is built incrementally by:

  • Starting with discrete points (0-cells)
  • Attaching n-dimensional cells along their boundaries to the (n-1)-skeleton
  • The n-skeleton consists of all cells of dimension ≤ n

CW complexes provide a way to decompose topological spaces into simple building blocks:

  • 0-cells: points
  • 1-cells: line segments
  • 2-cells: disks
  • 3-cells: solid balls, etc.

Structs§

Cell
A k-cell of arbitrary type and dimension
Skeleton
A Skeleton is a collection of Cells that have been glued together along Cell::attach maps.