o-minimal structures and definable functions Consider the following definition of an o-minimal structure:
An o-minimal structure $O=\{O_n\}$ is a sequence of Boolean algebras $O_n$ of subsets of $\mathbb{R}$ which satisfies the following axioms:


*

*$O$ is closed under cross products.

*$O$ is closed under axis-aligned projections $\mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$.

*$O$ contains all algebraic sets.

*$O_1$ consists of all finite unions of points and open intervals.


Now, we call the elements of $O$ definable, and we say that a function between  definable sets is definable if its graph is a definable set.
Now, the last condition, together with axiom 2 says that for a function to be definable, the image of any definable set is required to be definable. However, in other theories one usually requires the preimage of "nice" sets to be "nice", for the function to be "nice". For example, in measure theory one requires the preimage of a measurable set to be measurable, in topology one requires the preimage of an open set to be open, etc.
Why are definable functions in o-minimal structures defined by the "opposite" condition?
 A: The projections in condition 2 of your definition are used to model closure under existential quantification: if $A \in O_n$, then $\{(x_1, \ldots, x_n) \mathrel{|} \exists x_n \cdot (x_1, \ldots, x_{n-1}) \in A\} \in O_{n-1}$. In the motivating examples of o-minimal structures, like the theory of real closed fields, one has an algebraic definition of $O_n$ that does not involve quantifiers together with a quantifier-elimination theorem showing that the $O_n$ are closed under projections.
An o-minimal structure may be viewed as a tractable system for defining subsets of $\mathbb{R}^n$ (or $X^n$ for some other ordered set $X$). Under this view, lots of standard set-theoretic constructions preserve definability: definable relations have definable inverses, definable sets have definable images and pre-images under definable relations, the graphs of definable functions are definable, etc. Phenomena like a continuous bijection with a discontinuous inverse that arise in the category of topological spaces don't arise. The "oppositeness" is specific to the projections used to define o-minimality and don't reflect any asymmetries in the categories of definable sets and relations that the definitions lead to.
