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Are there any topological spaces $X$ with subspaces $A$ such that $H_n(X,A)$ is not isomorphic to $H_n(X/A)$?

I've been trying some familiar spaces, but everything seems to be me an isomorphism via the quotient map. Does anyone know of any examples?

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Hatcher gives the following as an excercise in section 2.1 of his Algebraic topology book. Let $X = [0,1]$ and $A = \{\frac{1}{n}\} \cup \{0\}$. Then $H_1(X,A)$ is not isomorphic to $H_1(X/A)$. –  Jason DeVito Nov 19 '12 at 17:59

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