Final topology

In topology and related areas of mathematics, the final topology on a set <math>X<math> is the strongest topology to make a family of functions into <math>X<math> continuous.

Definition

Given a set <math>X<math> and a familiy of topological spaces <math>(Y_i,\tau_i)<math> with functions

<math>f_i: Y_i \to X<math>

the final topology <math>\tau<math> on <math>X<math> is the strongest topology such that each

<math>f_i: (Y_i,\tau_i) \to (X,\tau)<math>

is continuous.

Examples

• The quotient topology is the final topology on the quotient space with respect to the quotient map.
• The direct sum topology is the final topology with respect to the family of canonical injections.

Properties

• A subset of <math>X<math> is open (closed) if and only if it is open (closed) in all <math>Y_i<math>.
• A function <math>g<math> from <math>X<math> to some space <math>Z<math> is continuous if and only if for each <math>f_i<math> <math>g \circ f_i<math> is continuous.

See also

• Initial topology