weak homotopy equivalence

A continuous map $f:X\to Y$ between path-connected basedtopological spaces is said to be a weak homotopy equivalence if for each $k\ge 1$ it induces an isomorphism $f_{*}:{\pi }_k(X)\to {\pi }_k(Y)$ between the$k$th homotopy groups. $X$ and $Y$ are then said to be weaklyhomotopy equivalent.

Remark 1.

It is not enough for ${\pi }_k\mathit{}\mathrm{(}X\mathrm{)}$ to be isomorphic to${\pi }_k\mathit{}\mathrm{(}Y\mathrm{)}$ for all $k\mathrm{.}$ The definition requires these isomorphismsto be induced by a space-level map $f\mathrm{.}$

Remark 2.

More generally, two spaces $X$ and $Y$ are defined to be weakly homotopy equivalent if there is a sequence of spaces and maps

$$X\to X_1\leftarrow X_2\to X_3\leftarrow \mathrm{\cdots }\to X_n\leftarrow Y$$

in which each map is a weak homotopy equivalence.