Hatcher states the following theorem on page 114 of his Algebraic Topology:
If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence $...\longrightarrow\widetilde{H}_n(A)\overset{i_*}\longrightarrow \widetilde{H}_n(X)\overset{j_*}\longrightarrow\widetilde{H}_n(X/A)\overset{\partial}\longrightarrow \widetilde{H}_{n-1}(A)\overset{i_*}\longrightarrow... $
where $i: A\hookrightarrow X$ is the inclusion and $j:X \rightarrow X/A$ is the quotient map.
Perhaps I am having a brain malfunction at the moment, but what are some interesting nonempty spaces which do not satisfy these criterion? By interesting, I mean something that appears "in nature."