Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$? Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$?
Motivation: I took this for granted for a long time, as I thought collapsing the contractible subspace does not change the homotopy type. Now it seems that this is only true for a CW pair... 

You are right. An interesting example of this kind of behavior consists of taking two copies of the Hawaiian Earring space and connecting their basepoints by a line segment. Contracting the middle segment gives you the standard Hawaiian Earrings. However this contraction is not a homotopy equivalence! The fundamental groups are different!
