Cellular filtration

Definition

A filtration of a topological space (say $X^n$ the filtration for $X$) is said to be a cellular filtration if it satisfes the following conditions:

• $H_p(X_n,X_{n-1})=0$for all $p\ne n$
• For any singular simplex, we can find a $n$ such that the image of the simplex sits inside $X^n$

A topological space equipped with the additional structure of a cellular filtration is termed a cellular space.

Relation with other structures

Stronger structures

• CW complex: The structure of a CW complex on a topological space automatically gives it a cellular filtration