There is an exercise in Bredon's "Topology and Geometry" in the Van Kampen section that supposes that $\mathbb{R}P^2=U_1\cup\cdots\cup U_n$ where each $U_i$ is open and homeomorphic to $\mathbb{R}^2$. The problem is to show that, for some $1\leq i\leq n$, we must have $(U_1\cup\cdots\cup U_{i-1})\cap U_i$ is either disconnected or empty. My assumption is that I need to use Van Kampen's Theorem.
My issue is the case where $n=2$. Then, for contradiction, I assume that $U_1\cap U_2$ is connected and non-empty. But, I can't apply Van Kampen because I only have that it is connected, not path connected.
Is there some fact that says if $X$ and $Y$ are homeomorphic to $\mathbb{R}^2$ then $X\cap Y$ is connected if and only if it is path connected? Or, is there some way of doing this problem with only connectedness? I can do the problem using homology.