Since , is closed. Also, and imply that , or is closed. If are closed, then is closed as well. Finally, suppose are closed. Let . For each , , so . This means , or . But by definition, so , or that is closed. ∎
so defined is called the closure topology of with respect to the closure operator .
Since the distributivity of over does not hold in general, and there is no guarantee that , a closure space under these generalized versions is a more general system than a topological space.
- 1 N. M. Martin, S. Pollard: Closure Spaces and Logic, Springer, (1996).
|Date of creation||2013-03-22 16:48:08|
|Last modified on||2013-03-22 16:48:08|
|Last modified by||CWoo (3771)|