Let $f$ be a continuous map of topological spaces $X \to Y$. Assume that $X$ and $Y$ are CW complexes. Let $X_+$ be the following space: take $X$ and attach a 0-cell (disjointly). Same for $Y_+$. Suppose the induced map $X_+ \to Y_+$ is a weak homotopy equivalence of pointed spaces. Does this mean that $f$ was a weak homotopy equivalence?
I'm getting confused on the definitions. If I have a random WHE $X_+ \to Y_+$, then it's definitely not true, but I couldn't think of a counterexample when the WHE is induced by a continuous map of unbased spaces.
The homotopy groups depend on the base point and they only depend on the path component this base point belongs to. If you choose base points in $X$ and $Y$, then the homotopy groups of $X_+$ and $Y_+$ with respect to this base point are the same as for $X$ and $Y$ because the added point belongs to another path component. In particular $f$ induces the same map between homotopy groups of $X_+$ and $Y_+$ as it does for those of $X$ and $Y$ and thus is again a weak homotopy equivalence.