Minimum number of sets required for a good open cover

A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible.

For example, $S^2$ has an open cover consisting of two sets: $S^2 \setminus \{northpole\}$ and $S^2 \setminus \{southpole\}$, however this is not a good open cover since the intersection is not contractible. Indeed, wikipedia claims that you need four sets to form a good open cover of $S^2$.

For a given topological space*, what is the minimum number of sets required for a good open cover?

*Of course, I understand that this question is probably way too broad to to say anything meaningful about, so I would be happy with partial, or qualified results. For instance, can we say anything in the case that $X$ is a surface? A manifold? A CW complex? A sphere, torus, or other relatively simple manifold?

Linked in the comments is a very relevant overflow post in which Mark Grant discusses the basic ideas surrounding the problem. Mark is not only a decent writer, but also a bit of an expert in these things, so you should read his post if you have not done so already.

Basically, there are a couple of issues surrounding the problem. Following Karoubi and Weibel we have the following definition.

The strict covering type of a space $X$ is the minimal cardinality of a good cover for it.

As mentioned, this was introduced originally by M. Karoubi and C. Weibel in On the covering type of a space, L’Ens. Math. 62, (2016), 457–474 (a free version is also available on Karoubi's homepage).

While this notion embraces the question, it has one major flaw: the strict covering type is not a homotopy invariant. This makes its computation unnecessarily difficult, as even in the simplest cases it will depend on such choices as the particular CW structure or triangulation with which a given space is equipped. For this reason Karoubi and Weibel also introduced a second quantity.

The covering type $ct(X)$ of a space $X$ is defined to be the minimal strict covering type of any space homotopy equivalent to $X$.

Here you should probably interpret 'space' as 'finite complex' to get a satisfactory theory. Of course the covering type of $X$ is manifestly homotopy invariant.

Unfortunately there isn't too much more that I can say at this point. Not all that much is really known about the covering types of even finite complexes. A thorough understanding of this quantity is a very open problem. Anyone interested in it should start by reading Karoubi-Weibel's paper, and from here consult the more recent works by Petar Pavešić et al (here, and here, say).

From this point on I'll write specifically to adress a question brought up by John Samples. Namely the question of the covering types of the torus, projective plane, and Klein bottle.

Already Karoubi-Weibel compute the covering type of the projective plane. In Theorem 6 of their paper they construct an explicit triangulation of $\mathbb{R}P^2$, and show that taking the open stars of its vertices yields $$ct(\mathbb{R}P^2)=6.$$

Karoubi-Weibel also compute the covering type of the torus in $\S5$ of their paper, where they show that $$ct(T^2)=6.$$ Again their method is to construct an explicit triangulation of $T^2$ and look at the family of its open stars. They show that the quantity they achieve in this manner is also an upper bound by using cohomological methods similar to those described by Zhen Lin's comment above.

They leave open the problem of the computation of the Klein bottle $N_2$, obtaining only upper and lower bounds. However $ct(N_2)$ is now known. It was obtained only recently as part of a much more general statement due to E. Borghini, and E. Minian, The covering type of closed surfaces and minimal triangulations, J. Comb Th, A, 166, (2019), 1-10 (arxiv version here).

They show:

[Gabriel, Minian] If $X$ is homotopy equivalent to a finite CW complex, then $ct(X)$ agrees with the minimum possible number of vertices of a simplicial complex homotopy equivalent to $X$.

This pleasant statement open up the computation of the covering type of surfaces to attack using classical information.

Theorem [Jungerman Ringel]: Let $S$ be a closed surface different from the orientable surface $M_2$ of genus $2$, the Klein bottle $N_2$, and the non-orientable surface $N_3$ of genus $3$. There exists a triangulation of $S$ with $n$ vertices if and only if $$n\geq\frac{7 +\sqrt{49−24\chi(S)}}{2}.$$ When $S$ is one of $M_2$, $N_2$, or $N_3$, then this formula applies when $n$ is replaced with $n−1$.

In particular this leads to the covering type of the Klein bottle as $$ct(N_2)=8.$$ See Gabriel-Minian's paper for a thorough discussion of this calculation.

I think these latter results lean more towards what John had in mind, and show that his idea expressed in the comments to compute $ct(T^2)$ is essentially correct. They also show that in the world of surfaces, that the covering type and strict covering type can be made to agree under a reasonable definition of surface.