If $\mathbb{R}^n=U\cup V$ for path-connected $U,V$ then $U\cap V$ is path-connected. Suppose that we can write
$$
\mathbb{R}^n=U\cup V,
$$
where $U,V$ are open and path-connected. It is easy to show using Mayer-Vietoris sequence that $U\cap V$ is path-connected as well.
Is there a proof which does not use homology, ideally something just using point-set topology?

This fact is a  Corollary of the many pointed version of the Seifert-van Kampen theorem, published in 1967.  
If $X= U \cup V$  is path connected, where $U,V$ are open and path connected, and  $U \cap V$ has $n$ path components,  then the fundamental group of $X$ has the free group $F$ on $n-1$ generators as a retract. See Topology and Groupoids Section 8.4;   see also this  paper on the  Phragmen-Brouwer Property,  which gives a small correction to that section, and  further reference to the literature. This paper proves by groupoid methods that if instead $U,V,W=U \cap V$ have $n_U,n_v,n_W$ path components, then the rank of the free group $F$ is $n_W-n_U-n_V+1$. 
