product topology. Endow the image with the final topology induced on it by the bijection that is, it is endowed with the Euclidean topology transferred to it from via This topology on is equal to the subspace topology induced on it by A subset is open (respectively, closed) in if and only if for every the set is an open (respectively, closed) subset of The topology is coherent with the family of subspaces This makes into an LB-space. Consequently, if and is a sequence in then in if and only if there exists some such that both and are contained in and in Often, for every the inclusion map is used to identify with its image in explicitly, the elements and are identified together. Under this identification, becomes a direct limit of the direct system where for every the map is the inclusion map defined by where there are trailing zeros. Categorical description In the language of category theory, the final topology construction can be described as follows. Let be a functor from a discrete category to the category of topological spaces Top that selects the spaces for Let be the diagonal functor from Top to the functor category TopJ (this functor sends each space to the constant functor to ). The comma category is then the category of co-cones from i.e. objects in are pairs where is a family of continuous maps to If is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to SetJ then the comma category is the category of all co-cones from The final topology construction can then be described as a functor from to This functor is left adjoint to the corresponding forgetful functor. See also Notes Citations References . (Provides a short, general introduction in section 9 and Exercise 9H) General topology