In general topology and related areas of mathematics, the final topology (or coinduced, weak, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that makes all those functions continuous. The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is coherent with some collection of subspaces if and only if it is the final topology induced by the natural inclusions. The dual notion is the initial topology, which for a given family of functions from a set X into topological spaces is the coarsest topology on X that makes those functions continuous.

Definition

Given a set X and an I-indexed family of topological spaces with associated functions the is the finest topology on X such that is continuous for each i\in I. Explicitly, the final topology may be described as follows: The closed subsets have an analogous characterization: The family \mathcal{F} of functions that induces the final topology on X is usually a set of functions. But the same construction can be performed if \mathcal{F} is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory. In that case there is always a subfamily \mathcal{G} of \mathcal{F} with \mathcal{G} a set, such that the final topologies on X induced by \mathcal{F} and by \mathcal{G} coincide. For more on this, see for example the discussion here. As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.

Examples

The important special case where the family of maps \mathcal{F} consists of a single surjective map can be completely characterized using the notion of quotient map. A surjective function between topological spaces is a quotient map if and only if the topology \tau on X coincides with the final topology induced by the family. In particular: the quotient topology is the final topology on the quotient space induced by the quotient map. The final topology on a set X induced by a family of X-valued maps can be viewed as a far reaching generalization of the quotient topology, where multiple maps may be used instead of just one and where these maps are not required to be surjections. Given topological spaces X_i, the disjoint union topology on the disjoint union is the final topology on the disjoint union induced by the natural injections. Given a family of topologies on a fixed set X, the final topology on X with respect to the identity maps as i ranges over I, call it \tau, is the infimum (or meet) of these topologies in the lattice of topologies on X. That is, the final topology \tau is equal to the intersection Given a topological space (X,\tau) and a family of subsets of X each having the subspace topology, the final topology induced by all the inclusion maps of the C_i into X is finer than (or equal to) the original topology \tau on X. The space X is called coherent with the family \mathcal C of subspaces if the final topology coincides with the original topology \tau. In that case, a subset will be open in X exactly when the intersection U\cap C_i is open in C_i for each i\in I. (See the coherent topology article for more details on this notion and more examples.) As a particular case, one of the notions of compactly generated space can be characterized as a certain coherent topology. The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms. Explicitly, this means that if is a direct system in the category Top of topological spaces and if is a direct limit of in the category Set of all sets, then by endowing X with the final topology induced by becomes the direct limit of in the category Top. The étalé space of a sheaf is topologized by a final topology. A first-countable Hausdorff space (X, \tau) is locally path-connected if and only if \tau is equal to the final topology on X induced by the set of all continuous maps where any such map is called a path in (X, \tau). If a Hausdorff locally convex topological vector space (X, \tau) is a Fréchet-Urysohn space then \tau is equal to the final topology on X induced by the set of all arcs in (X, \tau), which by definition are continuous paths that are also topological embeddings.

Properties

Characterization via continuous maps

Given functions from topological spaces Y_i to the set X, the final topology on X with respect to these functions f_i satisfies the following property: This property characterizes the final topology in the sense that if a topology on X satisfies the property above for all spaces Z and all functions g:X\to Z, then the topology on X is the final topology with respect to the f_i.

Behavior under composition

Suppose is a family of maps, and for every i \in I, the topology \upsilon_i on Y_i is the final topology induced by some family of maps valued in Y_i. Then the final topology on X induced by \mathcal{F} is equal to the final topology on X induced by the maps As a consequence: if is the final topology on X induced by the family and if is any surjective map valued in some topological space then is a quotient map if and only if (S, \sigma) has the final topology induced by the maps By the universal property of the disjoint union topology we know that given any family of continuous maps there is a unique continuous map that is compatible with the natural injections. If the family of maps f_i X (i.e. each x \in X lies in the image of some f_i) then the map f will be a quotient map if and only if X has the final topology induced by the maps f_i.

Effects of changing the family of maps

Throughout, let be a family of X-valued maps with each map being of the form and let denote the final topology on X induced by The definition of the final topology guarantees that for every index i, the map is continuous. For any subset the final topology on X will be than (and possibly equal to) the topology ; that is, implies where set equality might hold even if \mathcal{S} is a proper subset of If \tau is any topology on X such that and is continuous for every index i \in I, then \tau must be Comparison of topologies than (meaning that and this will be written ) and moreover, for any subset the topology \tau will also be than the final topology that \mathcal{S} induces on X (because ); that is, Suppose that in addition, is an A-indexed family of X-valued maps whose domains are topological spaces If every is continuous then adding these maps to the family \mathcal{F} will change the final topology on X; that is, Explicitly, this means that the final topology on X induced by the "extended family" is equal to the final topology induced by the original family However, had there instead existed even just one map g_{a_0} such that was continuous, then the final topology on X induced by the "extended family" would necessarily be Comparison of topologies than the final topology induced by that is, (see this footnote for an explanation).

Final topology on the direct limit of finite-dimensional Euclidean spaces

Let denote**** the , where *R*^{*N**}*** denotes the space of**** all real**** sequences.**** For every natural number n \in \N, let \R^n 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 \R^{\infty} with the final topology induced by the family of all inclusion maps. With this topology, \R^{\infty} 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 \R^{\infty} by \R^{\N}, where \R^{\N} is endowed with its usual 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 \R^n 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 n \in \N, 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 v_{\bull} is a sequence in \R^{\infty} then in if and only if there exists some n \in \N such that both v and v_{\bull} are contained in and in Often, for every n \in \N, the inclusion map is used to identify \R^n 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 m \leq n, the map is the inclusion map defined by where there are n - m trailing zeros.

Categorical description

In the language of category theory, the final topology construction can be described as follows. Let Y be a functor from a discrete category J to the category of topological spaces Top that selects the spaces Y_i for i \in J. Let \Delta be the diagonal functor from Top to the functor category TopJ (this functor sends each space X to the constant functor to X). The comma category is then the category of co-cones from Y, i.e. objects in are pairs (X, f) where is a family of continuous maps to X. If U 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 UY. The final topology construction can then be described as a functor from to This functor is left adjoint to the corresponding forgetful functor.

Citations