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 $$f_i, 0 \le i \le n - 1$$, 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 defining ingredient::weak homotopy equivalence of topological spaces.