Recall that an open cover $\mathfrak{U} = \{ U_\alpha \}$ of a manifold $M$ is called a *good cover* if all possible finite intersections $U_{\alpha_1} \cap ... \cap U_{\alpha_n}$ are contractible.

Question: What is the minimum number of open sets required for a good cover of the 2-dimensional torus?

The picture below provides a good cover of the torus (i.e. opposite sides of the parallelogram identified as usual) using 7 open sets (i.e. take sufficiently small open neighbourhoods of the hexagons).  Can one do any better than 7?  If not, how does one prove that 7 is optimal?

[![Good cover of the torus with 7 open sets (neighbourhoods of the hexagons)][1]][1]


  [1]: https://i.sstatic.net/IEnS7.png