Topological structure (topology)
open topology, respectively, closed topology
A collection , respectively , of subsets of a set , satisfying the following properties: 1) The set , as well as the empty set , are elements of the collection , respectively .
, respectively . The intersection, respectively union, of a finite number, and the union, respectively intersection, of any number of elements of , respectively , is an element of the same collection.
Once a topology, or topological structure, has been introduced or defined on a set , the set is called a topological space, its elements are called points and the elements of the collection , respectively , are called the open, respectively closed, sets of this topological space.
If one of the collections or of subsets of is defined, satisfying property 1 and or , respectively, then the other collection may be defined by duality as consisting of the complements of the elements of the first collection.
See also Topology, general; Topological space; General topology.
|[a1]||R. Engelking, "General topology" , Heldermann (1989)|
Topological structure (topology). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_structure_(topology)&oldid=18647