TITLE: If $A$ is a path connected subspace, is $q_*: \pi_1(X) \to \pi_1(X/A)$ a surjection?
QUESTION [5 upvotes]: Let $X$ be a topological space and $A \subset X$ be a path connected subspace. Let $q: X \to X/A$ be the quotient map. Every example i work out it seems that the induced map 
$q_*: \pi_1(X) \to \pi_1(X/A)$ is a surjection. Is this always true? Are there some conditions under which it is true?
Thank you in advance!

REPLY [5 votes]: For CW complexes, you can do the following:

Use the long exact sequence of homotopy groups to prove that $\pi_1(X)\rightarrow \pi_1(X,A)$ is surjective.
Use excision for homotopy groups (Hatcher, Prop. 4.28) to prove that $\pi_1(X,A)\rightarrow \pi_1(X/A)$ is surjective.

EDIT: There is a shorter argument using Van Kampen's theorem: $X/A$ is homotopy equivalent to the union of $X$ and $CA$ (the cone of $A$) along $A$. As all three spaces are path connected, we have that the homomorphism $\pi_1(X)*\pi_1(CA)\to \pi_1(X/A)$ is surjective, and since $CA$ is contractible, we have that $\pi_1(X)\to \pi_1(X/A)$ is surjective as well. Notice that this works in more generality than CW-complexes.