Initial topology
In topology and related areas of mathematics, the initial topology on a set is the weakest topology that makes one or more specified functions on that set continuous.
Let <math>f_i: X \to Y_i<math> be a family of functions, each from a fixed set <math>X<math> to a topological space <math>Y_i<math>. The initial topology on <math>X<math> (with respect to the family of functions) is the weakest topology such that every <math>f_i<math> is continuous. The topology is generated by sets of the form <math>f_i^{-1}(U)<math>, where <math>U<math> is an open set in <math>Y_i<math>.
Examples
Several topological constructions can be regarded as special cases of the initial topology.
- The subspace topology is the initial topology on the subspace with respect to the inclusion map.
- The product topology is the initial topology with respect to the family of projection maps.
- The weak topology is the inital topology on a locally convex space with respect to the continuous linear forms of its dual space
See also
Categories: Topology