Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$ be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$: $$ V\mapsto f'\mathcal{F}(V):=\varinjlim_{U\supseteq f(V)}\mathcal{F}(U) $$ One may check that $f'\mathcal{F}$ gives a presheaf on $X$. Under some strong assumptions one may show that $f'\mathcal{F}$ is a sheaf on $X$. Let me give two examples
If $Y$ is Hausdorff, locally compact and paracompact and $f$ is a closed embedding then $f'\mathcal{F}$ is actually a sheaf (this is not a trivial exercise, it took me a while to figure that out).
A variation of 1. is: if $Y$ is Hausdorff and locally compact, and $f$ is the embedding of a compact subset $Y\subseteq X$.
I got interested in this question since it is directly related to the proper base change theorem in topology which says the following:
Proper base change theorem: Let $f:X\rightarrow Y$ be a proper map with $Y$ Hausdorff and locally compact and $X$ paracompact. Then for any sheaf $\mathcal{F}$ on $Y$ and $y\in Y$ one has that \begin{align}\label{eqn} R^qf_*(\mathcal{F})_y\simeq H^q(X_y, f|_{X_y}^{-1}\mathcal{F}) \hspace{2cm} (\star) \end{align} where $X_y=f^{-1}(y)$ is the fiber above $y$.
So here are 3 questions:
Q 1 Is there a common generalization of 1. and 2. in the topological setting ?
Q 2 I would like to have a couple of (non-artificial ) examples where the presheaf $f'\mathcal{F}$ fails to be a sheaf in order to have a feeling for the possible geometrical (and/or topological) obstructions. (Note that this is closely related to examples of maps where the isomorphism $(\star)$ above fail).
Q 3 To what extend is it possible to generalize the proper base change theorem in the topological setting? (so here I have in mind of relaxing the assumptions on $f$ and may be adding additional restrictions on $Y$)
