Cellular filtration

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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}) = 0for 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