I'm going through Čech cohomology at a gentle pace and as you know we need a colimit to obtain the sought-after sheaf cohomology. In practice, for calculating actual cohomology groups, I would invoke Leray's theorem and choose an acyclic covering, so that there is no need for a colimit, and that's quite clear so far. On sunny days I also have a version of Mayer-Vietoris that makes computations tractable.
My issue is that I assumed quite naively that contractible spaces (say, open disks) have trivial cohomology (say, for the constant presheaf). And I get the expected results so it's not obviously wrong. But how can I prove this?
I'm aware that Čech cohomology is isomorphic to other cohomology theories for "nice enough" space, and that would be a way (albeit an impractical one) to get a proof. But surely there is a simple and elegant argument "internal" to Čech cohomology?