In Analysis, functions are often characterized as "structure-preserving" if structures from codomains are preserved into the domain under the preimage operation. Specifically, if we let $f: (A, S_1) \rightarrow (B, S_2)$ be a function s.t. $S_1 \subseteq P(A)$ and $S_2 \subseteq P(B)$, then we say that $f$ is "structure preserving" if for any arbitrary $s_2 \in S_2$ we have that $f^{-1}(s_2) \in S_1$. For example, we could let $S_1, S_2$ characterize the open subsets of $A,B$ and consider that the notion of "continuity" is that of "any open subset of $B$ has associated with it an open subset of $A$ under the pre-image operation". Alternatively, we could let $S_1$ and $S_2$ characterize measurable subsets of $A$ and $B$ respectively and consider that the notion of "measurable function" is characterized as "a function in which the preimage of a measurable subset of $B$ is itself measurable in $A$".
But why do we characterize the notion of "structure preserving" in terms of pre-images instead of images? Why not, for example, say that our function $f$ is structure preserving if for some $s_1 \in S_1$ we have that $f(s_1) \in S_2$? This to me seems more intuitive than the traditional approach.