Retraction and intersection

Question

Let $X$ be a topological space, and consider two open subsets $U$, $V$ of $X$ such that there exist two continuous maps $r_{U}: X\longrightarrow U$, $r_{V}:X\longrightarrow V$ which are homotopically equivalent to retractions. Assuming that $U\cap V\ne \emptyset$, is it possible to constuct a map $r_{U\cap V}: X\longrightarrow U\cap V$ which is homotopically equivalent to a retraction? If not, are there non-trivial additional hypothesis on the choice of $U$ and $V$ which would make things work?