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... 

Let me note a general fact: if the inclusion $A \hookrightarrow X$ (for $A$ a closed subspace) is a cofibration, and $A$ is contractible, then the map $X \to X/A$ is a homotopy equivalence. See Corollary 5.13 in chapter 1 of Whitehead's "Elements of homotopy theory."
