Weak homotopy-equivalent topological spaces

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Definition

Two topological spaces X_1 and X_2are said to be weak homotopy-equivalent or weakly homotopy-equivalent if the following is true:

We can find a collection of topological spaces X_1 = Y_0, Y_1, Y_2, \dots, Y_n = X_2 and a collection of continuous maps f_i, 0 \le i \le n - 1, such that each f_i is a map either from Y_i to Y_{i+1} or a map from Y_{i+1} to Y_i, and further, each f_i is a weak homotopy equivalence of topological spaces.

Relation with other equivalence relations

Stronger equivalence relations

Equivalence relation Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
homeomorphic spaces there is a bijective homeomorphism between the spaces
homotopy-equivalent topological spaces there is a homotopy equivalence of topological spaces between the spaces homotopy-equivalent not implies weak homotopy-equivalent

Weaker equivalence relations

Equivalence relation Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Spaces that have isomorphic homotopy groups isomorphic homotopy groups not implies weak homotopy-equivalent
homology-equivalent topological spaces generated by back and forth homology equivalences homology-equivalent not implies weak homotopy-equivalent
Spaces that have isomorphic homology groups and isomorphic fundamental groups and set of path components isomorphic homology groups and isomorphic fundamental groups not implies weak homotopy-equivalent