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Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
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Projection to trivial reduced cohomology class in $L^2(\mathbb{R})$

Given that I have had no success on the mathematics stackexchange (see here), I've decided to try my luck here. I am attempting to solve the following exercise (original formulation here), which to my ...
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