Final topology

Given a set X and an I-indexed family of topological spaces \left(Y_i, \upsilon_i\right) with associated functions f_i : Y_i \to X, the {{em|final topology on X induced by the family of functions \mathcal{F} := \left\{ f_i : i \in I \right\}}} is the finest topology \tau_{\mathcal{F}} on X such that f_i : \left(Y_i, \upsilon_i\right) \to \left(X, \tau_{\mathcal{F}}\right) is continuous for each i\in I. The final topology always exists, and is unique. Explicitly, the final topology may be described as follows: :a subset U of X is open in the final topology \left(X, \tau_{\mathcal{F}}\right) (that is, U \in \tau_{\mathcal{F}}) if and only if f_i^{-1}(U) is open in \left(Y_i, \upsilon_i\right) for each i\in I. The closed subsets have an analogous characterization: :a subset C of X is closed in the final topology \left(X, \tau_{\mathcal{F}}\right) if and only if f_i^{-1}(C) is closed in \left(Y_i, \upsilon_i\right) for each i\in I. 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 f : (Y, \upsilon) \to \left(X, \tau\right) between topological spaces is a quotient map if and only if the topology \tau on X coincides with the final topology \tau_{\mathcal{F}} induced by the family \mathcal{F}=\{f\}. 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 \coprod_i X_i is the final topology on the disjoint union induced by the natural injections. Given a family of topologies \left(\tau_i\right)_{i \in I} on a fixed set X, the final topology on X with respect to the identity maps \operatorname{id}_{\tau_i} : \left(X, \tau_i\right) \to X as i ranges over I, call it \tau, is the infimum (or meet) of these topologies \left(\tau_i\right)_{i \in I} in the lattice of topologies on X. That is, the final topology \tau is equal to the intersection \tau = \bigcap_{i \in I} \tau_i. Given a topological space (X,\tau) and a family \mathcal C=\{ C_i : i\in I\} of subsets of X each having the subspace topology, the final topology \tau_{\mathcal C} 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 \tau_{\mathcal C} coincides with the original topology \tau. In that case, a subset U\subseteq X 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 \operatorname{Sys}_Y = \left(Y_i, f_{ji}, I\right) is a direct system in the category Top of topological spaces and if \left(X, \left(f_i\right)_{i \in I}\right) is a direct limit of \operatorname{Sys}_Y in the category Set of all sets, then by endowing X with the final topology \tau_{\mathcal{F}} induced by \mathcal{F} := \left\{ f_i : i \in I \right\}, \left(\left(X, \tau_{\mathcal{F}}\right), \left(f_i\right)_{i \in I}\right) becomes the direct limit of \operatorname{Sys}_Y 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 C\left([0, 1]; X\right) of all continuous maps [0, 1] \to (X, \tau), 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 \operatorname{Arc}\left([0, 1]; X\right) of all arcs in (X, \tau), which by definition are continuous paths [0, 1] \to (X, \tau) that are also topological embeddings. == Properties ==

Characterization via continuous maps Given functions f_i : Y_i \to X, 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: :a function g from X to some space Z is continuous if and only if g \circ f_i is continuous for each i \in I. 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 \mathcal{F} := \left\{ f_i:Y_i \to X \mid i \in I \right\} 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 \mathcal{G}_i 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 \left\{ f_i \circ g ~:~ i \in I \text{ and } g \in \cal G_i \right\}. As a consequence: if \tau_{\mathcal{F}} is the final topology on X induced by the family \mathcal{F} := \left\{ f_i : i \in I \right\} and if \pi : X \to (S, \sigma) is any surjective map valued in some topological space (S, \sigma), then \pi : \left(X, \tau_{\mathcal{F}}\right) \to (S, \sigma) is a quotient map if and only if (S, \sigma) has the final topology induced by the maps \left\{ \pi \circ f_i ~:~ i \in I \right\}. By the universal property of the disjoint union topology we know that given any family of continuous maps f_i : Y_i \to X, there is a unique continuous map f : \coprod_i Y_i \to X 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 \mathcal{F} := \left\{ f_i : i \in I \right\} be a family of X-valued maps with each map being of the form f_i : \left(Y_i, \upsilon_i\right) \to X and let \tau_{\mathcal{F}} denote the final topology on X induced by \mathcal{F}. The definition of the final topology guarantees that for every index i, the map f_i : \left(Y_i, \upsilon_i\right) \to \left(X, \tau_{\mathcal{F}}\right) is continuous. For any subset \mathcal{S} \subseteq \mathcal{F}, the final topology \tau_{\mathcal{S}} on X will be than (and possibly equal to) the topology \tau_{\mathcal{F}}; that is, \mathcal{S} \subseteq \mathcal{F} implies \tau_{\mathcal{F}} \subseteq \tau_{\mathcal{S}}, where set equality might hold even if \mathcal{S} is a proper subset of \mathcal{F}. If \tau is any topology on X such that \tau \neq \tau_{\mathcal{F}} and f_i : \left(Y_i, \upsilon_i\right) \to (X, \tau) is continuous for every index i \in I, then \tau must be Comparison of topologies| than \tau_{\mathcal{F}} (meaning that \tau \subseteq \tau_{\mathcal{F}} and \tau \neq \tau_{\mathcal{F}}; this will be written \tau \subsetneq \tau_{\mathcal{F}}) and moreover, for any subset \mathcal{S} \subseteq \mathcal{F} the topology \tau will also be than the final topology \tau_{\mathcal{S}} that \mathcal{S} induces on X (because \tau_{\mathcal{F}} \subseteq \tau_{\mathcal{S}}); that is, \tau \subsetneq \tau_{\mathcal{S}}. Suppose that in addition, \mathcal{G} := \left\{g_a : a \in A\right\} is an A-indexed family of X-valued maps g_a : Z_a \to X whose domains are topological spaces \left(Z_a, \zeta_a\right). If every g_a : \left(Z_a, \zeta_a\right) \to \left(X, \tau_{\mathcal{F}}\right) is continuous then adding these maps to the family \mathcal{F} will change the final topology on X; that is, \tau_{\mathcal{F} \cup \mathcal{G}} = \tau_{\mathcal{F}}. Explicitly, this means that the final topology on X induced by the "extended family" \mathcal{F} \cup \mathcal{G} is equal to the final topology \tau_{\mathcal{F}} induced by the original family \mathcal{F} = \left\{ f_i : i \in I \right\}. However, had there instead existed even just one map g_{a_0} such that g_{a_0} : \left(Z_{a_0}, \zeta_{a_0}\right) \to \left(X, \tau_{\mathcal{F}}\right) was continuous, then the final topology \tau_{\mathcal{F} \cup \mathcal{G}} on X induced by the "extended family" \mathcal{F} \cup \mathcal{G} would necessarily be Comparison of topologies| than the final topology \tau_{\mathcal{F}} induced by \mathcal{F}; that is, \tau_{\mathcal{F} \cup \mathcal{G}} \subsetneq \tau_{\mathcal{F}} (see this footnote for an explanation). == Final topology on the direct limit of finite-dimensional Euclidean spaces ==

Let \R^{\infty} ~:=~ \left\{ \left(x_1, x_2, \ldots\right) \in \R^{\N} ~:~ \text{ all but finitely many } x_i \text{ are equal to } 0 \right\}, 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 \operatorname{In}_{\R^n} : \R^n \to \R^{\infty} denote the inclusion map defined by \operatorname{In}_{\R^n}\left(x_1, \ldots, x_n\right) := \left(x_1, \ldots, x_n, 0, 0, \ldots\right) so that its image is \operatorname{Im} \left(\operatorname{In}_{\R^n}\right) = \left\{ \left(x_1, \ldots, x_n, 0, 0, \ldots\right) ~:~ x_1, \ldots, x_n \in \R \right\} = \R^n \times \left\{ (0, 0, \ldots) \right\} and consequently, \R^{\infty} = \bigcup_{n \in \N} \operatorname{Im} \left(\operatorname{In}_{\R^n}\right). Endow the set \R^{\infty} with the final topology \tau^{\infty} induced by the family \mathcal{F} := \left\{ \; \operatorname{In}_{\R^n} ~:~ n \in \N \; \right\} 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 \tau^{\infty} 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 \operatorname{Im} \left(\operatorname{In}_{\R^n}\right) with the final topology induced on it by the bijection \operatorname{In}_{\R^n} : \R^n \to \operatorname{Im} \left(\operatorname{In}_{\R^n}\right); that is, it is endowed with the Euclidean topology transferred to it from \R^n via \operatorname{In}_{\R^n}. This topology on \operatorname{Im} \left( \operatorname{In}_{\R^n} \right) is equal to the subspace topology induced on it by \left(\R^{\infty}, \tau^{\infty}\right). A subset S \subseteq \R^{\infty} is open (respectively, closed) in \left(\R^{\infty}, \tau^{\infty}\right) if and only if for every n \in \N, the set S \cap \operatorname{Im} \left(\operatorname{In}_{\R^n}\right) is an open (respectively, closed) subset of \operatorname{Im} \left(\operatorname{In}_{\R^n}\right). The topology \tau^{\infty} is coherent with the family of subspaces \mathbb{S} := \left\{ \; \operatorname{Im} \left(\operatorname{In}_{\R^n}\right) ~:~ n \in \N \; \right\}. This makes \left(\R^{\infty}, \tau^{\infty}\right) into an LB-space. Consequently, if v \in \R^{\infty} and v_{\bull} is a sequence in \R^{\infty} then v_{\bull} \to v in \left(\R^{\infty}, \tau^{\infty}\right) if and only if there exists some n \in \N such that both v and v_{\bull} are contained in \operatorname{Im} \left(\operatorname{In}_{\R^n}\right) and v_{\bull} \to v in \operatorname{Im} \left(\operatorname{In}_{\R^n}\right). Often, for every n \in \N, the inclusion map \operatorname{In}_{\R^n} is used to identify \R^n with its image \operatorname{Im} \left(\operatorname{In}_{\R^n}\right) in \R^{\infty}; explicitly, the elements \left( x_1, \ldots, x_n \right) \in \R^n and \left(x_1, \ldots, x_n, 0, 0, 0, \ldots\right) are identified together. Under this identification, \left(\left(\R^{\infty}, \tau^{\infty}\right), \left(\operatorname{In}_{\R^n}\right)_{n \in \N}\right) becomes a direct limit of the direct system \left(\left(\R^n\right)_{n \in \N}, \left(\operatorname{In}_{\R^m}^{\R^n}\right)_{m \leq n \text{ in } \N}, \N\right), where for every m \leq n, the map \operatorname{In}_{\R^m}^{\R^n} : \R^m \to \R^n is the inclusion map defined by \operatorname{In}_{\R^m}^{\R^n}\left(x_1, \ldots, x_m\right) := \left(x_1, \ldots, x_m, 0, \ldots, 0\right), 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 (Y \,\downarrow\, \Delta) is then the category of co-cones from Y, i.e. objects in (Y \,\downarrow\, \Delta) are pairs (X, f) where f = (f_i : Y_i \to X)_{i \in J} 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 \left(UY \,\downarrow\, \Delta^{\prime}\right) is the category of all co-cones from UY. The final topology construction can then be described as a functor from \left(UY \,\downarrow\, \Delta^{\prime}\right) to (Y \,\downarrow\, \Delta). This functor is left adjoint to the corresponding forgetful functor. == See also ==