the family of subspaces if where denotes the final topology induced by the inclusion maps where for every the inclusion map takes the form Unraveling the definition, is coherent with if and only if the following statement is true: for every subset is open in if and only if for every is open in the subspace Closed sets can be checked instead: is coherent with if and only if for every subset is closed in if and only if for every is closed in For example, if is a cover of a topological space by open subspaces (i.e. open subsets of endowed with the subspace topology) then is coherent with In contrast, if is the set of all singleton subsets of (each set being endowed with its unique topology) then is coherent with if and only if is the discrete topology on The disjoint union is the final topology with respect to the family of canonical injections. A space is called and a if is coherent with the set of all compact subspaces of All first-countable spaces and all Hausdorff locally compact spaces are -spaces, so that in particular, every manifold and every metrizable space is coherent with the family of all its compact subspaces. As demonstrated by the examples that follows, under certain circumstance, it may be possible to characterize a more general final topology in terms of coherence with subspaces. Let be a family of -valued maps with each map being of the form and let denote the final topology on induced by Suppose that is a topology on and for every index the image is endowed with the subspace topology inherited from If for every the map is a quotient map then if and only if is coherent with the set of all images Final topology on the direct limit of finite-dimensional Euclidean spaces Let denote the , where denotes the space of all real sequences. For every natural number let denote the usual Euclidean space endowed with the Euclidean topology and let denote the inclusion map defined by so that its image is and consequently, Endow the set with the final topology induced by the family of all inclusion maps. With this topology, becomes a complete Hausdorff locally convex sequential topological vector space that is a Fréchet–Urysohn space. The topology is strictly finer than the subspace topology induced on by where is endowed with its usual