preservation and reflection

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 transformationsPlanetmathPlanetmath 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 setMathworldPlanetmath. 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 spaceMathworldPlanetmath is preserved under a continuous functionPlanetmathPlanetmath. Here, the constructs are topological spaces, and the transformation is a continuous function. In other words, if f:XY is a continuous function from X to Y. If X is connected, so is f(X)Y.

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

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 theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath, and is almost exclusively reserved for functorsMathworldPlanetmath. For example, a faithful functorMathworldPlanetmath reflects isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath: if F is a faithful functor from 𝒞 to 𝒟, and the object F(A) is isomorphic to the object F(B) in 𝒟, then A is isomorphic to B in 𝒞.

Title preservation and reflection
Canonical name PreservationAndReflection
Date of creation 2013-03-22 17:12:18
Last modified on 2013-03-22 17:12:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 00A35
Defines preserve
Defines reflect