# Soft sheaves on $T_1$-space

Suppose we have a topological space $$X$$ and an open subset $$U \subseteq X$$ with inclusion $$j \colon U \hookrightarrow X$$. If we have a sheaf $$\mathcal F \in \text{Sh}(X)$$, then we know that in general the canonical sheaf morphism $$\mathcal F \to j_*\mathcal F|_U$$ is not surjective. It is, though, for flasque sheaves.

I was wondering if the surjectivity still holds if $$\mathcal F$$ is only soft, but $$X$$ is $$T_1$$, i.e. every single-point-set is closed. I think the answer is yes, because as $$\{x\}$$ is closed for every $$x \in X$$, the softness of $$\mathcal F$$ implies the surjectivity of the canonical map $$\mathcal F(X) \to \mathcal F_x$$ which also implies the surjectivity of $$\mathcal F_x \to (j_*\mathcal F|_U)_x.$$

Is this correct or am I missing something? Thanks in advance for any help / comments!