Let be a topological space. A subset of is a basis for if every member of is a union of members of .
Equivalently, is a basis if and only if whenever is open and then there is an open set such that .
The topology generated by a basis consists of exactly the unions of the elements of .
We also have the following easy characterization: (for a proof, see the attachment)
1. A basis for the usual topology of the real line is given by the set of open intervals since every open set can be expressed as a union of open intervals. One may choose a smaller set as a basis. For instance, the set of all open intervals with rational endpoints and the set of all intervals whose length is a power of are also bases. However, the set of all open intervals of length is not a basis although it is a subbasis (since any interval of length less than can be expressed as an intersection of two intervals of length ).
2. More generally, the set of open balls forms a basis for the topology on a metric space.
3. The set of all subsets with one element forms a basis for the discrete topology on any set.
|Date of creation||2013-03-22 12:05:03|
|Last modified on||2013-03-22 12:05:03|
|Last modified by||rspuzio (6075)|
|Synonym||topology generated by a basis|