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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 $H_1(X/A)$.2012-11-19

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Here is an extremely simple example:
Let $X=U^2\subset \mathbb R^2$ be the open unit disc with center the origin $O$ and let $A=U\setminus \{O\}$.
The long exact sequence for the pair $(X,A)$ yields the segment $\cdots \to H_2(X)=0 \to H_2(X,A) \to H_1 (A)=\mathbb Z \to H_1(X)=0 \to\cdots $ the displayed equalities following from $X$ and $A$ being homotopic respectively to a point and a circle. So we obtain from that segment

First result : $H_2(X,A)=\mathbb Z$

On the other hand the quotient space $X/A$ is the Sierpinski space, a two element space with one closed and one open point. Such a space is known to be contractible, so that we now obtain

Second result: $H_2(X/A)=0$

Conclusion: The pair $(X=U^2,A=U\setminus \{O\})$ is an example for which the relative homology does not coincide with the homology of the quotient space.