In mathematics, the word ``preserve'' usually means the ``preservation of properties''. Loosely speaking, whenever a mathematical construct $A$ has some property $P$ , after $A$ is somehow ``transformed'' into $A'$ , the transformed object $A'$ also has property $P$ . The constructs usually refer to sets and the transformations typically are functions or something similar.

Here is a simple example, let $f$ be a function from a set $A$ to $B$ . Let $A$ be a finite set. Let $P$ be the property of a set being finite. Then $f$ preserves $P$ , since $f(A)$ is finite. Note that we are not saying that $B$ is finite. We are merely saying that the portion of $B$ that is the image of $A$ (the transformed portion) is finite.

Here is another example. The property of being connected in a topological space is preserved under a continuous function. Here, the constructs are topological spaces, and the transformation is a continuous function. In other words, if $f:X\to Y$ is a continuous function from $X$ to $Y$ . If $X$ is connected, so is $f(X)\subseteq Y$ .

Many more examples can be found in abstract algebra. Group homomorphisms, for example, preserve commutativity, as well as the property of being finitely generated.

The word ``reflect'' is the dual notion of ``preserve''. It means that if the transformed object has property $P$ , then the original object also has property $P$ . This usage is rarely found outside of category theory, and is almost exclusively reserved for functors. For example, a faithful functor reflects isomorphism: if $F$ is a faithful functor from $\mathcal{C}$ to $\mathcal{D}$ , and the object $F(A)$ is isomorphic to the object $F(B)$ in $\mathcal{D}$ , then $A$ is isomorphic to $B$ in $\mathcal{C}$ .