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D'oh, I missed the possibility of an open domain. Thanks!

And, even better, the dimension of the kernel of d just counts how many connected components the domain has.

There is a similar thing in graph theory, where the kernel of the incidence matrix counts how many connected components a graph has.

Yea, the statement is missing "locally constant". It's a poor example given the assumptions are not clearly stated.

Oh, these kind of problems happen time and again in real-life maths: you use a lemma until someone points out that you are assuming something which may not take place (like connectedness of the domain, here).

Yea, but like I said, it's a poor example as a "false belief" in this context, because as stated, it isn't false for some unwritten assumptions and is false for other unwritten assumptions. In mathematics, well and actually everything, the importance of assumptions is paramount. The problem is just ill-stated.

This happens in engineering and software all the time. Time and time again, issues being resolved usually revolves around clarifying assumptions.