Let be a topological vector space whose continuous dual separates points (i.e., is T2). The weak topology on is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of remains continuous on . To differentiate the topologies and , is sometimes referred to as the strong topology on .
Note that the weak topology is a special case of a more general concept. In particular, given a nonempty family of mappings from a set to a topological space , one can define a topology to be the collection of all unions and finite intersections of sets of the form with and an open set in . The topology -often called the -topology on and/or the weak topology on induced by -is the coarsest topology in which every element is continuous on and so it follows that the above-stated definition corresponds to the special case of for a topological vector space.