Editing Weak topology (section)

=== Weak topology induced by the continuous dual space ===

Alternatively, the '''weak topology''' on a TVS {{mvar|X}} is the [[initial topology]] with respect to the family <math>X^*</math>. In other words, it is the [[comparison of topologies|coarsest]] topology on X such that each element of <math>X^*</math> remains a [[continuous function]].

A [[subbase]] for the weak topology is the collection of sets of the form <math>\phi^{-1}(U)</math> where <math>\phi\in X^*</math> and {{mvar|U}} is an open subset of the base field <math>\mathbb{K}</math>. In other words, a subset of {{mvar|X}} is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form <math>\phi^{-1}(U)</math>.

From this point of view, the weak topology is the coarsest [[polar topology]].