Given a topological space $X$. A sheaf $\mathcal{F}$ on $X$ is flasque if for any open set $U\subseteq X$, $\Gamma(X,\mathcal{F})\to\Gamma(U,\mathcal{F})$ is surjective. And a sheaf $\mathcal{G}$ is soft if for any closed set $K\subseteq X$, $\Gamma(X,\mathcal{G})\to\Gamma(K,\mathcal{G})$ is surjective. It's said that flasque sheaf is obviously soft. But I am not quite sure, so I try to give a proof as follows.
Given an closed set $K\subseteq X$, and denote the boundary of $K$ as $\overline{K}$, then $K-\overline{K}$ is an open set, so $\mathcal{F}(X)\to\mathcal{F}(K-\overline{K})$ is surjective. We can cover the boundary $\overline{K}$ with open covers {$U_i$}, so for each {$U_i$}, we have $\mathcal{F}(X)\to\mathcal{F}(U_i)$ is surjective. Since $\mathcal{F}$ is a sheaf, we can make sure $\mathcal{F}(U_i)$ coincide with $\mathcal{F}(K-\overline{K})$ on $U_i \cap (K-\overline{K})$, so that we can glue. Then since {$U_i$} contain $\overline{K}$, we can actually extend $\mathcal{F}(K-\overline{K})$ to $\mathcal{F}(K)$ in $\mathcal{F}(X)$. So it's surjective for closed subset. Thus is soft.
Is my argument right? Any comment would help. Thanks!
I don't quite follow your argument, but in my opinion it is not at all obvious that flabby/flasque sheaves are soft.
First of all, you should write exactly how you define the sections of a sheaf over a closed subset.
Second, the implication flabby$\implies$soft relies on subtle assumptions on the topological space involving paracompactness, since the proof uses partitions of unity to glue locally defined sections.