preservation and reflection
In mathematics, the word “preserve” usually means the “preservation of properties”. Loosely speaking, whenever a mathematical construct has some property , after is somehow “transformed” into , the transformed object also has property . The constructs usually refer to sets and the transformations typically are functions or something similar.
Here is a simple example, let be a function from a set to . Let be a finite set. Let be the property of a set being finite. Then preserves , since is finite. Note that we are not saying that is finite. We are merely saying that the portion of that is the image of (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 is a continuous function from to . If is connected, so is .
The word “reflect” is the dual notion of “preserve”. It means that if the transformed object has property , then the original object also has property . This usage is rarely found outside of category theory, and is almost exclusively reserved for functors. For example, a faithful functor reflects isomorphism: if is a faithful functor from to , and the object is isomorphic to the object in , then is isomorphic to in .
|Title||preservation and reflection|
|Date of creation||2013-03-22 17:12:18|
|Last modified on||2013-03-22 17:12:18|
|Last modified by||CWoo (3771)|