Topological structure (topology)
open topology, respectively, closed topology
A collection , respectively \mathfrak F, of subsets of a set X, satisfying the following properties:
1. The set X, as well as the empty set \emptyset, are elements of the collection \mathfrak G, respectively \mathfrak F.
2_\mathfrak G, respectively 2_\mathfrak F. The intersection, respectively union, of a finite number, and the union, respectively intersection, of any number of elements of \mathfrak G, respectively \mathfrak F, is an element of the same collection.
Once a topology, or topological structure, has been introduced or defined on a set X, the set is called a topological space, its elements are called points and the elements of the collection \mathfrak G, respectively \mathfrak F, are called the open, respectively closed, sets of this topological space.
If one of the collections \mathfrak G or \mathfrak F of subsets of X is defined, satisfying property 1 and 2_\mathfrak G or 2_\mathfrak F, respectively, then the other collection may be defined by duality as consisting of the complements of the elements of the first collection.
Comments
See also Topology, general; Topological space; General topology.
References
[a1] | R. Engelking, "General topology" , Heldermann (1989) |
Topological structure (topology). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_structure_(topology)&oldid=33605