Decomposition of a space as a union of two closed subspaces that dosen't form an Mayer-Vetoris pair.

Question

When applying Mayer-Vietoris sequence in homology theory, one case is where $X=X_1\cup X_2$, $X_1$ and $X_2$ are closed subspaces of $X$ and $X_1\cap X_2$ is a deformation retract of one of its open neighborhood. I want to find examples where $X_i$ are closed subspaces of $X,X_1\cap X_2\neq \emptyset$ but don't form an Mayer-Vetoris pair. Appreciation for any comment!

Answer 1

Try Griffith's twin cone.

It is a union of two closed contractible subspaces which meet in a single point. Its first homology group is infinite.