Weak homotopy-equivalent topological spaces

Definition

Two topological spaces $X_{1}$ and $X_{2}$are 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 $f_{i},0\leq i\leq n-1$, 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