Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$.

My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal F_i$ also flabby?

Here is the difficulty:

Given an open subset $U\subset X$ a section $s\in \Gamma(U,\mathcal F)$ consists in a collection of sections $s_i\in \Gamma(U,\mathcal F_i)$ subject to the condition that for any $x\in U$ there exists a neighbourhood $x\in V\subset U$ on which almost all $s_i\vert V \in \Gamma(V,\mathcal F_i)$ are zero.

Now, every $s_i$ certainly extends to a section $S_i\in \Gamma(X,\mathcal F_i)$ by the flabbiness of $\mathcal F_i$.

The problem is that I see no reason why the collection $(S_i)_{i\in I}$ should be a section in $\Gamma(X,\oplus _{i\in I} \mathcal F_i)$, since I see no reason why every point in $X$ should have a neighbourhood $W$ on which almost all the restrictions $S_i\vert W$ are zero.

Of course any direct sum of flabby sheaves *is* flabby if the space $X$ is noetherian, since in that case we have $\Gamma(U,\mathcal F) =\oplus_{i\in I} \Gamma(U,\mathcal F_i)$ for all open subsets $U\subset X$.

I have only seen the fact that direct sums of flabby sheaves are flabby (correctly) used on noetherian spaces, actually schemes, so that my question originates just from idle curiosity...