If you allow the open sets $X$, $U$ and $V$ to be disconnected, you get a counter-example by taking a countable example and changing points to pairwise disjoint balls.

**Edit:** Because I still don't see how Joel's example works, I decided to modify it by drilling infinitely many holes (windows) to the house. Still the same shrinking by affine mapping of everything but the chimneys is performed to the house, but now the chimneys are mapped so that they fill in the space above the top window. The set $U$ is everything left from the blue line and $V$ is symmetrically from the other side.



<img src="http://users.jyu.fi/~tamaraja/temp/house.gif">