Hi everyone!

Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are a variety of definitions of soft sheaf, let me emphasize my definition (taken from Gelfand-Manin, Methods of Homological Algebra, p.38): $\mathcal{F}$ is soft if for any closed $Z \subset X$, the restriction map $$\mathcal{F}(X) \rightarrow \varinjlim_{U \supset Z} \mathcal{F}(U)$$is a surjection.

For an open embedding $j: U \rightarrow X$, is it true that $j^* \mathcal{F}$ is again soft?

Thanks so much!

What I understand/motivation: I can prove this when $U$ is again paracompact - while (Hausdorff) and (Hausdorff + locally compact) both are inherited by open subsets, my impression was that paracompactness does not always descend, hence my question. Also, to be fair Gelfand-Manin also ask that $X$ be separable - will this affect the answer?

By the way, the motivation for my question is in trying to understand their proof of Verdier duality; on p. 231 they state that if $L$ is soft, then so is $L \otimes j_!j^*\mathbb{Z}$. My approach would be $$L \otimes j_! j^* \mathbb{Z} \simeq j_! j^* L$$ and Gelfand-Manin state in an exercise (which I'm still puzzling out) that $f_!$ between locally compact Hausdorff spaces sends soft sheaves to soft sheaves.