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general topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from
topological spaces In mathematics, a topological space is, roughly speaking, a geometry, geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a Set (mathematics), set of ...
into X, is the finest topology on X that makes all those functions
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. 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 In general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used ...
is the final topology with respect to the
inclusion map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. The final topology is also the topology that every
direct limit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
in the
category of topological spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is Coherent topology, coherent with some collection of Subspace topology, 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 \left(Y_i, \upsilon_i\right) with associated functions f_i : Y_i \to X, the is the finest topology \tau_ on X such that f_i : \left(Y_i, \upsilon_i\right) \to \left(X, \tau_\right) is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
for each i\in I. Explicitly, the final topology may be described as follows: :a subset U of X is open in the final topology \left(X, \tau_\right) (that is, U \in \tau_) if and only if f_i^(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_\right) if and only if f_i^(C) is closed in \left(Y_i, \upsilon_i\right) for each i\in I.


Examples

The important special case where the family of maps \mathcal consists of a single surjective map can be completely characterized using the notion of Quotient map, quotient maps. 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_ induced by the family \mathcal=\. In particular: the quotient topology is the final topology on the quotient space induced by the Quotient space (topology)#Quotient map, 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), 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 sets, family of topologies \left(\tau_i\right)_ on a fixed set X, the final topology on X with respect to the identity maps \operatorname_ : \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)_ in the lattice of topologies on X. That is, the final topology \tau is equal to the Intersection (set theory), intersection \tau = \bigcap_ \tau_i. The
direct limit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of any Direct system (mathematics), 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_Y = \left(Y_i, f_, I\right) is a direct system in the Category of topological spaces, category Top of topological spaces and if \left(X, \left(f_i\right)_\right) is a
direct limit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of \operatorname_Y in the Category of sets, category Set of all sets, then by endowing X with the final topology \tau_ induced by \mathcal := \left\, \left(\left(X, \tau_\right), \left(f_i\right)_\right) becomes the direct limit of \operatorname_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 (mathematics), path in (X, \tau). If a Hausdorff Locally convex topological vector space, 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\left([0, 1]; X\right) of all Arc (topology), arcs in (X, \tau), which by definition are continuous Path (mathematics), 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 can be characterized by 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.


Behavior under composition

Suppose \mathcal := \left\ 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_i of maps valued in Y_i. Then the final topology on X induced by \mathcal is equal to the final topology on X induced by the maps \left\. As a consequence: if \tau_ is the final topology on X induced by the family \mathcal := \left\ and if \pi : X \to (S, \sigma) is any surjective map valued in some topological space (S, \sigma), then \pi : \left(X, \tau_\right) \to (S, \sigma) is a quotient map if and only if (S, \sigma) has the final topology induced by the maps \left\. 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 := \left\ 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_ denote the final topology on X induced by \mathcal. 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_\right) is continuous. For any subset \mathcal \subseteq \mathcal, the final topology \tau_ on X will be Comparison of topologies, than (and possibly equal to) the topology \tau_; that is, \mathcal \subseteq \mathcal implies \tau_ \subseteq \tau_, where set equality might hold even if \mathcal is a proper subset of \mathcal. If \tau is any topology on X such that for all i \in I, f_i : \left(Y_i, \upsilon_i\right) \to (X, \tau) is continuous but \tau \neq \tau_, then \tau is Comparison of topologies, then \tau_ (in symbols, \tau \subsetneq \tau_ which means \tau \subseteq \tau_ and \tau \neq \tau_) and moreover, for any subset \mathcal \subseteq \mathcal, because \tau_ \subseteq \tau_, the topology \tau will also be than the final topology \tau_ induced on X by \mathcal; that is \tau \subsetneq \tau_. Suppose that in addition, \mathcal := \left\ is a family of X-valued maps 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_\right) is continuous then adding these maps to the family \mathcal will change the final topology on X; that is, \tau_ = \tau_. Explicitly, this means that the final topology on X induced by the "extended family" \mathcal \cup \mathcal is equal to the final topology \tau_ induced by the original family \mathcal = \left\. However, had there instead existed even just one map g_ such that g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right) was continuous, then the final topology \tau_ on X induced by the "extended family" \mathcal \cup \mathcal would necessarily be Comparison of topologies, than the final topology \tau_ induced by \mathcal; that is, \tau_ \subsetneq \tau_ (see this footnoteBy definition, the map g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right) not being continuous means that there exists at least one open set U \in \tau_ such that g_^(U) is not open in \left(Z_, \zeta_\right). In contrast, by definition of the final topology \tau_, the map g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right) be continuous. So the reason why \tau_ must be strictly coarser, rather than strictly finer, than \tau_ is because the failure of the map g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right) to be continuous necessitates that one or more open subsets of \tau_ must be "removed" in order for g_ to become continuous. Thus \tau_ is just \tau_ but some open sets "removed" from \tau_. for an explanation).


Coherence with subspaces

Let (X, \tau) be a topological space and let \mathbb be a Family of sets, family of subspaces of (X, \tau) where importantly, the word "sub" is used to indicate that each subset S \in \mathbb is endowed with the subspace topology \tau\vert_S inherited from (X, \tau). The space (X, \tau) is said to be with the family \mathbb of subspaces if \tau = \tau_, where \tau_ denotes the final topology induced by the
inclusion map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s \mathcal := \left\ where for every S \in \mathbb, the inclusion map takes the form :\operatorname_S^X : \left( S, \tau\vert_S \right) \to X. Unraveling the definition, (X, \tau) is coherent with \mathbb if and only if the following statement is true: :for every subset U \subseteq X, U is open in (X, \tau) if and only if for every S \in \mathbb, U \cap S is open in the Subspace topology, subspace \left(S, \tau\vert_S\right). Closed sets can be checked instead: (X, \tau) is coherent with \mathbb if and only if for every subset C \subseteq X, C is closed in (X, \tau) if and only if for every S \in \mathbb, C \cap S is closed in \left( S, \tau\vert_S \right). For example, if \mathbb is a cover of a topological space (X, \tau) by open subspaces (i.e. open subsets of (X, \tau) endowed with the subspace topology) then \tau is coherent with \mathbb. In contrast, if \mathbb is the set of all Singleton set, singleton subsets of (X, \tau) (each set being endowed with its unique topology) then (X, \tau) is coherent with \mathbb if and only if \tau is the discrete topology on X. The Disjoint union (topology), disjoint union is the final topology with respect to the family of canonical injections. A space (X, \tau) is called and a if \tau is coherent with the set \mathbb of all compact subspaces of (X, \tau). All first-countable spaces and all Hausdorff space, 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 \mathcal := \left\ 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_ denote the final topology on X induced by \mathcal. Suppose that \tau is a topology on X and for every index i \in I, the Image (mathematics), image \operatorname f_i := f_i(X) is endowed with the subspace topology \tau\vert_ inherited from (X, \tau). If for every i \in I, the map f_i : \left(Y_i, \upsilon_i\right) \to \left( \operatorname f_i, \tau\vert_ \right) is a quotient map then \tau = \tau_ if and only if (X, \tau) is coherent with the set of all images \left\.


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

Let \R^ ~:=~ \left\, denote the , where \R^ 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_ : \R^n \to \R^ denote the
inclusion map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
defined by \operatorname_\left(x_1, \ldots, x_n\right) := \left(x_1, \ldots, x_n, 0, 0, \ldots\right) so that its Image (mathematics), image is \operatorname \left(\operatorname_\right) = \left\ = \R^n \times \left\ and consequently, \R^ = \bigcup_ \operatorname \left(\operatorname_\right). Endow the set \R^ with the final topology \tau^ induced by the family \mathcal := \left\ of all inclusion maps. With this topology, \R^ becomes a Complete topological vector space, complete Hausdorff space, Hausdorff Locally convex topological vector space, locally convex Sequential space, sequential topological vector space that is a Fréchet–Urysohn space. The topology \tau^ is Comparison of topologies, strictly finer than the subspace topology induced on \R^ by \R^, where \R^ is endowed with its usual product topology. Endow the image \operatorname \left(\operatorname_\right) with the final topology induced on it by the bijection \operatorname_ : \R^n \to \operatorname \left(\operatorname_\right); that is, it is endowed with the Euclidean topology transferred to it from \R^n via \operatorname_. This topology on \operatorname \left( \operatorname_ \right) is equal to the subspace topology induced on it by \left(\R^, \tau^\right). A subset S \subseteq \R^ is open (resp. closed) in \left(\R^, \tau^\right) if and only if for every n \in \N, the set S \cap \operatorname \left( \operatorname_ \right) is an open (resp. closed) subset of \operatorname \left( \operatorname_ \right). The topology \tau^ is coherent with family of subspaces \mathbb := \left\. This makes \left(\R^, \tau^\right) into an LB-space. Consequently, if v \in \R^ and v_ is a sequence in \R^ then v_ \to v in \left(\R^, \tau^\right) if and only if there exists some n \in \N such that both v and v_ are contained in \operatorname \left(\operatorname_\right) and v_ \to v in \operatorname \left(\operatorname_\right). Often, for every n \in \N, the inclusion map \operatorname_ is used to identify \R^n with its image \operatorname \left(\operatorname_\right) in \R^; explicitly, the elements \left( x_1, \ldots, x_n \right) \in \mathbb^n and \left(x_1, \ldots, x_n, 0, 0, 0, \ldots\right) are identified together. Under this identification, \left(\left(\R^, \tau^\right), \left(\operatorname_\right)_\right) becomes a
direct limit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the direct system \left(\left(\R^n\right)_, \left(\operatorname_^\right)_, \N\right), where for every m \leq n, the map \operatorname_^ : \R^m \to \R^n is the inclusion map defined by \operatorname_^\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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
Top that selects the spaces Y_i for i \in J. Let \Delta be the diagonal functor from Top to the functor category Top''J'' (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_i : Y_i \to X is a family of continuous maps to X. If Y is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set''J'' then the comma category \left(UY \,\downarrow\, \Delta^\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^\right) to (Y \,\downarrow\, \Delta). This functor is Adjoint functors, left adjoint to the corresponding forgetful functor.


See also

* * * * *


Notes


Citations


References

* . ''(Provides a short, general introduction in section 9 and Exercise 9H)'' * General topology