Intersection preserves homotopy equivalence

Question

Let $Z$ be a topological space with subspaces $X$, $Y$, $X'$ and $Y'$. Suppose that $X$ is homotopy equivalent to $X'$ and $Y$ is homotopy equivalent to $Y'$ do we have that $X\cap Y$ is homotopy equivalent to $X'\cap Y'$? thanks for your help!

Answer 1

No, there's no reason for this to be true. Take everything to be a subspace of $\mathbb{R}^2$ and $Y = Y'$ to both be the $x$-axis (this is to make the example as easy to visualize as possible). Now take $X$ to be a circle intersecting the $x$-axis at two points and $X'$ to be a circle tangent to the $x$-axis. (Of course by deforming the circle we can arrange for even more intersection points.)