TITLE: Are the fundamental groups of $X$ and $X/A$ isomorphic when $A$ is contractible?
QUESTION [5 upvotes]: Let $X$ be a topological space, $A$ a contractible subspace of $X$, and $f : X \rightarrow X/A$ the quotient map. I want to say that $f$ always induces an isomorphism between fundamental groups, or at least that it does under certain assumptions (e.g., when $X$ is a CW complex), but I do not see how to show this explicitly except when $X$ is a finite graph and $A$ is a tree in $X$. Is there a way to prove this in general, or at least in a more general case than finite graphs?

REPLY [7 votes]: The result isn't true in general.
Take $X=S^1$, $A=S^1 -\{N\}$, $N$ being the north pole. $A$ is contractible and $X/A$ is the Sierpiński space, which is contractible (and thus has a trivial fundamental group), but $X$ has a non-trivial fundamental group.