So I would need to a kind of measure!

@DavidGao Oh I get what you say. By density I meant some thing as follows: Let A is a subset of of topological measur space the density of A at a point p is the limit $\frac{\mu (A\cap U)}{\mu(U)}$ where U shrink to p among open neighborhoods of p

@DavidGao No the density is obvious but the 2 items I mentioned is my main questions.

@DavidGao Obviously both $A$ and $S\setminus A$ are dense. I do not get what do you meant?

Thank you for your comment. To be honnest my comment is not significant and is not worth of an answer. I will think a little more and come back here with possible elaboration on your interesting question. I belive that an algebraic formulation of your question is an interesting problem

In your question are you also interested to consider the contravariant case $\ell:X\to Y$ but $h:\mathcal{G} to \mathcal{F}$? In this case every continuous function $\ell:X\to Y$ gives a natural pull back $h:=\ell^*: C(Y) \to C(X)$ with the property you mentioned. I like your question because it can be applied to every possible category

Do I understand the definition of SIP correctly?: If $N_1$ and $N_2$ are complemented submodules of M then their intersection is a complemented submodule again

With a Non commutative point of view whose world consits of point less space they are not rare: every commutative von neumann algebra is an example of an extremely disconnected space: Consider its spectrum