Finite cell complex

Definition

space, map.
The space “obtained by attaching an -cell to along ” is:
is the smallest equivalence relation matching for all .

Intuition

We basically just glue the boundary of some disc to .
Easiest to see for because there we just pick two points
(not necessarily different) in and attach a lil path between them.

Definition

A finite cell complex (cx) is a space obtained by

1. Start with finite set with discrete topology. (0-skeleton)
2. Iteratively: -skeleton is obtained from -skeleton by attaching finitely many -cells.
3. Stop after finitely many steps

Intuition

Start with finitely many points.
Then attach little paths between them (kinda like a graph now)
Then attach circles along those paths.
You could either:

• attach a circle by folding it over
• or you could attach it normally along a loop
• or you could probably do other things
  Then attach balls along those circles
  A bit harder to visualize, but for example
  if you had a fully enclosed section you could put a ball inside there to fill it up
  etc