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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.

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It is a manifestation of the duality between algebra and geometry: preservation of structure in the sense of your last paragraph is typical in algebra, but in geometry where we are more interested in "nearness" rather than "equality" it makes more sense for preservation properties to go the other way. (Just think about the definition of continuity!) – Zhen Lin Dec 4 '12 at 14:11

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