Let be a nonempty set and be a family of subsets of . is a topology on if it satisfies the following properties:
(a) and
(b) Any union of elements in is also an element of
(c) Any intersection of finitely many elements of is an element of
is a topological space, and the members of are open sets in
is closed if is open.
Function respects boolean operations.
Relative Topology
Let be a topological space, and a nonempty subset of . Define the relative topology, . Then we can we define open, closed, and compactness relative to . A neighborhood of is any open set containing . is a topology.
is the discrete topology. All subsets of are closed and open