Cover of $S^1$ such that their intersections are path connected

Question

I thinking about this problem when I try to compute the fundamental group of $S^1$ using Seifert-van Kampen Theorem.

There is a cover of at least two path connected opens $U_{\alpha}$ of $S^1$ such that All pairwise intersection are non-empty and path connected?

I think that such cover is impossible. But, I don't know how proof this.