Point-set topology is a fundamental branch of topology, sometimes referred to as general topology, which deals with the concepts of topological spaces and the mathematical structures defined on such spaces.
Topology and open sets[]
Given a set , a family of subsets of is said to be a topology of if the following three conditions hold:
- (The empty set and are both elements of )
- (Any union of elements of is an element )
- (Any finite intersection of elements of is an element of )
The members of a topology are called open sets of the topology.
Topological space[]
A topological space is a set , known as the underlying set, together with a topology T of .
Basis for a topology[]
A basis for a topology on is a collection of subsets of , known as basis elements, such that the following two properties hold:
- For every there is at least one basis element that contains .
- If is an element of the intersection of two basis elements , then there exists a basis element such that .
Given a basis for a topology, one can define the topology generated by the basis as the collection of all sets such that for each there is a basis element such that and .
Closed sets[]
A set is defined to be closed if its complement in is an open set in the given topology.
Neighborhoods[]
A set is said to be a neighborhood of a point if it is an open set which contains the point . In some cases the term neighborhood is used to describe a set which contains an open set containing .
Interior and closure[]
The interior of a subset of is defined to be the union of all open sets contained in .
The closure of a subset of is defined as the intersection of all closed sets containing .
Limit points[]
A point of is said to be a limit point of a subset A of if every neighborhood of intersects A in at least one point other than .
Continuous functions[]
A function is said to be continuous if for each subset of , the set is an open set of .
Homeomorphisms[]
A bijective function is said to be a homeomorphism if both and its inverse, , are continuous.
If there exists a homeomorphism between two topological spaces X and Y, then the spaces are said to be homeomorphic.
Any property that is invariant under homeomorphisms is known as a topological property.
A homeomorphism is also dubbed a topological equivalence among mathematicians.