union of two contractible spaces, having nonempty path-connected intersection, need not be contractible

Question

show that union of two contractible spaces, having nonempty path-connected intersection, need not be contractible.

can someone give me a proper example please.I could not remind anything.

Answer 1

Consider sphere $S^2$ with two open subsets $U,V$, s.t. $U$ contains everything but the south pole, $V$ contains everything but the north pole. They are both contractible, their intersection is homotopic to the circle, which is path connected, but their union is $S^2$, not contractible.