Fact-checked by Grok 2 weeks ago

Final topology

In , the final topology on a set Y induced by a family of continuous maps f_i: (X_i, \tau_i) \to Y for i \in I, where each (X_i, \tau_i) is a , is defined as the collection \sigma = \{V \subseteq Y \mid f_i^{-1}(V) \in \tau_i \ \forall i \in I\}; this is the finest (largest) topology on Y such that every f_i is continuous. A key feature of the final topology is its universal property: for any topological space (Z, \eta) and map g: Y \to Z, g is continuous with respect to \sigma on Y if and only if the compositions g \circ f_i: X_i \to Z are continuous for all i \in I. This property ensures that the final topology is uniquely determined as the strongest topology compatible with the given maps, distinguishing it from coarser topologies that might also make the f_i continuous but fail the universal condition. Notable examples include the quotient topology, where the final topology on the quotient set X / \sim with respect to the canonical projection \pi: X \to X / \sim consists of sets W \subseteq X / \sim such that \pi^{-1}(W) is open in X; this construction is fundamental in identifying points under equivalence relations while preserving continuity. Another example is the sum topology (or disjoint union topology) on \bigoplus_{i \in I} X_i, induced as the final topology with respect to the inclusion maps i_j: X_j \hookrightarrow \bigoplus_{i \in I} X_i, where open sets are unions of opens from the component spaces. These constructions highlight the final topology's role in categorical colimits within the category of topological spaces.

Fundamentals

Definition

In , a map f: Y \to X between a (Y, \tau_Y) and a set X is continuous with respect to a \tau on X if the preimage f^{-1}(U) is open in Y for every U \in \tau. Given a set X and a family of maps \{f_i: Y_i \to X \mid i \in I\}, where each Y_i is a , the final on X induced by this family is the finest \tau on X such that every f_i is continuous. This consists precisely of those subsets U \subseteq X for which f_i^{-1}(U) is open in Y_i for every i \in I. The final topology exists and is unique; it is the finest topology on X that renders each f_i continuous, obtained as the intersection of the individual final topologies induced by each f_i. This construction is dual to the on a set, which is the coarsest topology making a of maps from the set to topological spaces continuous.

Comparison to Initial Topology

The initial topology on a set X with respect to a family of maps \{g_j: X \to Z_j\}_{j \in J} into topological spaces Z_j is defined as the coarsest topology on X that renders all the g_j continuous; its subbasis consists of the preimages \{g_j^{-1}(V_j) \mid V_j \text{ open in } Z_j, j \in J\}. In contrast to the final topology, which imposes the finest topology on a set X to make a given family of maps \{f_i: Y_i \to X\}_{i \in I} from topological spaces Y_i continuous—effectively pulling back open sets from the codomains to X—the initial topology pushes forward structure from the domain X via the maps to the codomains, generating the coarsest compatible structure. This duality positions the final topology as the supremum and the initial topology as the infimum in the lattice of all topologies on the underlying set, ordered by inclusion. Both concepts were formalized in the 1940s as part of the structural approach to in Nicolas Bourbaki's , particularly in the Topologie générale volumes, where the final topology plays a central role in constructing colimits; the core definitions and duality have remained foundational without significant revisions since.

Examples

Quotient and Disjoint Union Topologies

The topology arises as a prototypical instance of the final topology. Consider a Y and a surjective continuous q: Y \to X, where X is equipped with no prior topology. The final topology on X with respect to the single map q is precisely the topology, defined such that a U \subseteq X is open if and only if its preimage q^{-1}(U) is open in Y. This construction ensures that q is continuous and that the topology on X is the finest possible with this property. A notable application of the quotient topology as a final topology is the formation of the real projective plane \mathbb{RP}^2. This space is obtained by taking the 2-sphere S^2 with its standard topology and quotienting by the equivalence relation that identifies each point with its antipodal point, via the surjective map q: S^2 \to \mathbb{RP}^2 sending x to the line through the origin and x. The resulting quotient topology on \mathbb{RP}^2 makes it a compact, non-orientable manifold. The disjoint union topology similarly exemplifies the final topology in the context of coproducts. For an indexed family of topological spaces \{Y_i\}_{i \in I}, form the set-theoretic disjoint union X = \coprod_{i \in I} Y_i and consider the inclusion maps f_i: Y_i \to X. The final topology on X with respect to the family \{f_i\}_{i \in I} is the disjoint union topology, wherein a subset U \subseteq X is open if and only if U \cap Y_i is open in Y_i for every i \in I. This topology renders X the categorical coproduct of the Y_i in the category of topological spaces, with open sets consisting of arbitrary unions of open sets from the component spaces. When the maps f_i: Y_i \to X are inclusions of a family of subspaces \{Y_i\}_{i \in I} \subseteq X, each Y_i equipped with its own topology \tau_i, the final topology on X with respect to these inclusions is the coherent topology relative to the . Here, a U \subseteq X is open precisely when U \cap Y_i is open in (Y_i, \tau_i) for all i \in I. If the \{Y_i\} X, this yields the finest on X compatible with the given topologies on the subspaces, ensuring compatibility across the cover while inducing each \tau_i as the subspace topology on Y_i.

Direct Limits of Spaces

In the , the direct limit of a directed system (Y_\alpha, f_{\alpha\beta})_{\alpha \in A} consists of the set-theoretic colimit X = \varinjlim Y_\alpha equipped with the final induced by the canonical bonding maps \iota_\alpha: Y_\alpha \to X. This ensures that each \iota_\alpha is continuous and is the finest such on X, making the direct limit functorial in the . For sequential directed systems, such as increasing unions Y_1 \subseteq Y_2 \subseteq \cdots, a U \subseteq X is open U \cap Y_n is open in Y_n for every n. A concrete example arises from the directed system of finite sets \{ \{1, \dots, n\} \}_{n \in \mathbb{N}}, each endowed with the topology, connected by inclusion maps. The set-theoretic colimit is the \mathbb{N}, and the final coincides with the topology on \mathbb{N}, as every intersects each in either the or itself, both open. This illustrates how the final topology preserves discreteness in infinite colimits of spaces. In the context of topological vector spaces, the inductive limit topology—equivalent to the final topology with respect to the inclusions—is central to constructing spaces like LF-spaces, which are countable strict inductive limits of Fréchet spaces. Such topologies ensure that the resulting space is complete and locally convex, with the final topology guaranteeing continuity of the structure maps while inheriting desirable properties like barrelledness from the approximating spaces. In algebraic geometry, ind-schemes are direct limits of schemes, where the underlying topological space of \operatorname{Spec}( \varinjlim R_\alpha ) receives the final topology with respect to the maps \operatorname{Spec}(R_\beta) \to \operatorname{Spec}( \varinjlim R_\alpha ), facilitating the study of infinite-dimensional geometric objects.

Properties

Characterization via Continuous Maps

The final topology on a set X induced by a family of continuous maps f_i: Y_i \to X, where each Y_i is equipped with a topology, is uniquely characterized by its universal property with respect to continuous maps out of X. Specifically, for any topological space Z, a map h: X \to Z is continuous if and only if the compositions h \circ f_i: Y_i \to Z are continuous for every index i. This criterion ensures that the final topology is the finest topology on X making all the f_i continuous, as any coarser topology would fail to preserve the continuity of some h satisfying the condition. A subset U \subseteq X is open in the final topology if and only if its preimage f_i^{-1}(U) is open in Y_i for every i. Dually, a subset V \subseteq X is closed if and only if f_i^{-1}(V) is closed in Y_i for every i. These open and closed set tests reflect the colimit-like property of the final topology, where the structure on X is determined solely by pulling back the topologies from the Y_i via the inducing maps. To see the equivalence between this definition and the continuity preservation, consider that the final topology is generated by the subbasis consisting of sets of the form f_i(O_i), where O_i is open in Y_i. A map h: X \to Z is continuous precisely when it maps these subbasis elements to open sets in Z, which holds h \circ f_i is continuous on each Y_i, as the preimages under h align with those under the compositions. This subbasis argument confirms the without requiring a full of the topology axioms.

Stability under Operations

The final topology exhibits notable stability under various operations, ensuring that modifications to the underlying family of maps or subsequent mappings preserve structural properties in a predictable manner. Consider a family of maps \{f_i : Y_i \to X\}_{i \in I}, where each Y_i is equipped with a , inducing the final \tau on the set X. This \tau is the finest one making all f_i continuous. Now suppose there is an additional map g : X \to [W](/page/W). The final on W induced by the composed family \{g \circ f_i : Y_i \to W\}_{i \in I} coincides precisely with the on W obtained by pushing forward \tau via g. This property underscores the compatibility of the final with post-composition, as the continuity conditions for the compositions g \circ f_i translate directly to the structure on W. Regarding extensions of the family, the final topology is sensitive to changes in the indexing set I. If a new map f_j : Y_j \to X is added to the family, the resulting final topology on X becomes coarser than \tau, because the collection of subsets U \subseteq X for which all preimages f_i^{-1}(U) (now including f_j^{-1}(U)) are open in their respective domains is a subset of the previous collection, yielding fewer open sets. Conversely, removing a map from the family produces a finer topology, as fewer continuity constraints allow for more potential open sets. The topology remains unchanged under equivalent families, meaning those that generate the same subbasis consisting of images f_i(V_i) for open V_i \subseteq Y_i. This monotonicity with respect to family modifications highlights the robustness of the final topology as a minimal structure satisfying the continuity requirements. Subspace inheritance further illustrates stability. For a S \subseteq X, the on S induced from (X, \tau) is exactly the final topology on S with respect to the restricted family \{f_i|_{f_i^{-1}(S)} : f_i^{-1}(S) \to S\}_{i \in I}, where each restricted domain f_i^{-1}(S) inherits its topology from Y_i. This ensures that the relative structure on subspaces preserves the continuity properties of the original family, without introducing extraneous open sets.

Applications

Direct Limit of Finite-Dimensional Euclidean Spaces

The space \mathbb{R}^\infty, consisting of all real sequences with only finitely many nonzero terms, is constructed as the of the directed system (\mathbb{R}^n)_{n \in \mathbb{N}} in the of topological vector spaces, where the maps i_{n,m}: \mathbb{R}^n \to \mathbb{R}^m for n \leq m are the linear inclusions that embed \mathbb{R}^n into the first n coordinates of \mathbb{R}^m and set the remaining coordinates to zero. The final topology on \mathbb{R}^\infty is the unique finest topology relative to which all the canonical inclusion maps i_n: \mathbb{R}^n \to \mathbb{R}^\infty are continuous, ensuring that \mathbb{R}^\infty inherits the structure of an inductive limit in the . This final topology renders \mathbb{R}^\infty a complete, Hausdorff, locally topological , as it is a strict inductive limit of the Fréchet spaces \mathbb{R}^n equipped with their topologies. Although \mathbb{R}^\infty is not normable due to its infinite dimensionality and lack of metrizability, the topology restricted to any bounded subset coincides with the of the finite-dimensional subspace containing it, making such subsets metrizable. In this topology, a subset U \subseteq \mathbb{R}^\infty is open if and only if i_n^{-1}(U) is open in \mathbb{R}^n for every n \in \mathbb{N}. The neighborhoods of the origin thus consist of absorbing sets that are unions of the images i_n(V_n) under the inclusions, where each V_n is a convex neighborhood of the origin in \mathbb{R}^n. A sequence (x_k) in \mathbb{R}^\infty converges to a limit x if and only if there exists some n such that all but finitely many x_k and x lie in the image i_n(\mathbb{R}^n) and the sequence converges to x in the Euclidean topology of \mathbb{R}^n. The space \mathbb{R}^\infty finds applications in distribution theory, where spaces of test functions, such as those with compact support, are endowed with analogous inductive limit topologies to define distributions as continuous linear functionals. It also serves as a model for infinite-dimensional manifolds, enabling the study of smooth structures and in infinite dimensions.

Role in Categorical Colimits

In the category \mathbf{Top} of topological spaces and continuous maps, colimits are realized by first computing the underlying colimit in the category \mathbf{Set} of sets and functions, and then equipping the resulting set with the final topology relative to the canonical structure maps from each object in the diagram to the colimit. This ensures that the colimit object in \mathbf{Top} satisfies the universal property: any cocone of continuous maps from the diagram factors uniquely through the colimit via continuous maps. The final topology guarantees that these canonical maps are continuous quotients or inclusions, making the construction compatible with the category's morphisms. Specific examples illustrate this role. For pushouts, the colimit of a diagram X \leftarrow Z \to Y is the quotient of the disjoint union X \sqcup Y by the equivalence relation identifying points via the maps from Z, equipped with the quotient topology, which coincides with the final topology on the pushout. Similarly, coequalizers of parallel maps f, g: X \rightrightarrows Y are formed as the quotient set Y / \sim, where \sim is the equivalence generated by f(x) \sim g(x) for all x \in X, endowed with the quotient topology to ensure the coequalizing map is continuous. The U: \mathbf{Top} \to \mathbf{Set} commutes with colimits, meaning the underlying set of a colimit in \mathbf{Top} is the colimit in \mathbf{Set}, but the induced topology in \mathbf{Top} is the final one, which need not match the often associated with sets. For instance, a in \mathbf{Top} may carry the indiscrete topology if the maps identify all points, whereas embedding the set colimit discretely into \mathbf{Top} would yield a ; this discrepancy is particularly evident for finite sets, where the final is discrete only if the structure maps preserve separations.

Categorical Perspective

Universal Property

The final topology on a set X induced by a family of continuous maps \{f_i : Y_i \to X \mid i \in I\}, where each Y_i is a topological space, endows X with the finest topology \tau_{\text{final}} such that all f_i are continuous. This topology is defined such that a subset U \subseteq X is open if and only if f_i^{-1}(U) is open in Y_i for every i \in I. Categorically, the pair ((X, \tau_{\text{final}}), \{f_i\}_{i \in I}) satisfies a universal mapping property: it represents the functor F : \mathbf{Top}^{\text{op}} \to \mathbf{Set} that sends a topological space Z to the set \prod_{i \in I} \mathbf{Hom}_{\mathbf{Top}}(Y_i, Z) of families of continuous maps from the Y_i to Z. Specifically, \mathbf{Hom}_{\mathbf{Top}}((X, \tau_{\text{final}}), Z) \cong F(Z) naturally in Z, where the isomorphism sends a continuous map h : (X, \tau_{\text{final}}) \to Z to the family \{h \circ f_i \mid i \in I\}, and the inverse constructs the unique set-theoretic map from X to Z induced by the family (assuming the f_i are jointly surjective in \mathbf{Set}), which is continuous with respect to \tau_{\text{final}}. This representability implies that ((X, \tau_{\text{final}}), \{f_i\}_{i \in I}) is the initial object in the comma category (\{f_i\}_{i \in I} \downarrow \mathbf{[Top](/page/Top)}), whose objects are triples (Z, \{g_i : Y_i \to Z\}_{i \in I}, h : X \to Z) with each g_i continuous and g_i = h \circ f_i as set maps, and whose morphisms are continuous maps \phi : Z \to Z' commuting with the g_i and h. The initiality ensures that for any such object, there exists a unique continuous map \phi : (X, \tau_{\text{final}}) \to Z such that \phi \circ f_i = g_i for all i. This Yoneda-like highlights the final topology as the "free" completion of the underlying set to a continuous in \mathbf{[Top](/page/Top)}. The concept of the final topology and its emerged in the 1960s within , particularly through the development of topological functors and concrete categories by Horst Herrlich.

Relation to Functors and Limits

The final topology admits a functorial description in . Consider the category Fam(Top, Set), whose objects consist of a set S together with a family of continuous maps \{f_i: X_i \to S\}_{i \in I} from topological spaces X_i to S, and whose morphisms are pairs of set maps and commuting families of continuous maps. The \mathrm{Fin}: \mathrm{Fam}(\mathrm{Top}, \mathrm{Set}) \to \mathrm{Top} assigns to each such object the topological space (S, \tau_{\mathrm{final}}) equipped with the final topology relative to the family \{f_i\}, and acts on morphisms by the underlying set maps. This preserves colimits when restricted to appropriate slice categories, reflecting the colimit-preserving nature of final topologies in \mathrm{Top}. A fundamental duality exists between final topologies and within the \mathrm{[Top](/page/Top)} of topological spaces. The on a set with respect to a source family of maps from it to other spaces computes limits in \mathrm{[Top](/page/Top)}, such as products, by endowing the product set with the coarsest topology making all maps continuous. Dually, the final topology computes colimits in \mathrm{[Top](/page/Top)}, such as coproducts, by endowing the set with the finest topology making all maps continuous. This duality underscores how final topologies "push forward" structures from domain spaces to codomains, mirroring the "" role of . From the perspective of comma categories, the space endowed with the final topology relative to a \{f_i: X_i \to S\} serves as the initial object in the comma category (\Delta_S \downarrow \mathrm{[Top](/page/Top)}), where \Delta_S denotes the constant functor sending the terminal category to the underlying set S viewed in \mathrm{Set}, and morphisms in the comma category consist of continuous maps from S to other topological spaces that render the maps continuous after composition. This contrasts with the topology, which realizes the terminal object in the oppositely oriented comma category (\mathrm{[Top](/page/Top)} \downarrow \Delta^S). Such constructions highlight the universal properties governing topological colimits.