the category of Hausdorff topological spaces). Direct systems Suppose that is a directed set and that for all indices there are (continuous) morphisms in such that if then is the identity map on and if then the following compatibility condition is satisfied: where this means that the composition If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set is known as a direct system in the category that is directed (or indexed) by . Since the indexing set is a directed set, the direct system is said to be directed. The maps are called the bonding, connecting, or linking maps of the system. If the indexing set is understood then is often omitted from the above tuple (i.e. not written); the same is true for the bonding maps if they are understood. Consequently, one often sees written " is a direct system" where "" actually represents a triple with the bonding maps and indexing set either defined elsewhere (e.g. canonical bonding maps, such as natural inclusions) or else the bonding maps are merely assumed to exist but there is no need to assign symbols to them (e.g. the bonding maps are not needed to state a theorem). Direct limit of a direct system For the construction of a direct limit of a general inductive system, please see the article: direct limit. Direct limits of injective systems If each of the bonding maps is injective then the system is called injective. If the 's have an algebraic structure, say addition for example, then for any , we pick any index such that and then define their sum using by using the addition operator of . That is, where is the addition operator of . This sum is independent of the index that is chosen. In the category of locally convex topological vector spaces, the topology on the direct limit of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset of is a neighborhood of if and only if is an absolutely convex neighborhood of in for every index . Direct limits in Top Direct limits of directed direct systems always exist in the categories of sets, topological spaces, groups, and locally convex TVSs. In the category of topological spaces, if every bonding map is/is a injective (resp. surjective, bijective, homeomorphism,