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.

Stronger structures

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