Question on Proposition 17.5 in John Lee's Smooth Manifolds
Here is the proposition, for reference: Let ${M_j}$ be a countable collection of smooth $n$-manifolds with or without boundary, and let $M=∐_j M_j$. For each $p$, the inclusion maps $\iota_j :M_j ↪M$ induce an isomorphism from $H_p^{dR}(M)$ to the direct product space $∏_j H_p^{dR}(M_j)$.
I have two questions: why do they have to be countable? This is never used or mentioned in the proof. Also, why is it from $H_p^{dR}(M)$ to $∏_j H_p^{dR}(M_j)$? Shouldn't it be the other way around? Thanks!
Because by definition, a smooth manifold has to be 2nd countable.