Initial topology

Last updated

In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set , with respect to a family of functions on , is the coarsest topology on X that makes those functions continuous.


The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.

The dual notion is the final topology, which for a given family of functions mapping to a set is the finest topology on that makes those functions continuous.


Given a set X and an indexed family (Yi)iI of topological spaces with functions

the initial topology on is the coarsest topology on X such that each

is continuous.

Explicitly, the initial topology is the collection of open sets generated by all sets of the form , where is an open set in for some iI, under finite intersections and arbitrary unions. The sets are often called cylinder sets. If I contains exactly one element, all the open sets of are cylinder sets.


Several topological constructions can be regarded as special cases of the initial topology.


Characteristic property

The initial topology on X can be characterized by the following characteristic property:
A function from some space to is continuous if and only if is continuous for each i  I.

Characteristic property of the initial topology InitialTopology-01.png
Characteristic property of the initial topology

Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.


By the universal property of the product topology, we know that any family of continuous maps determines a unique continuous map

This map is known as the evaluation map.

A family of maps is said to separate points in X if for all in X there exists some i such that . Clearly, the family separates points if and only if the associated evaluation map f is injective.

The evaluation map f will be a topological embedding if and only if X has the initial topology determined by the maps and this family of maps separates points in X.

Separating points from closed sets

If a space X comes equipped with a topology, it is often useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient (but not necessary) condition.

A family of maps {fi: XYi} separates points from closed sets in X if for all closed sets A in X and all x not in A, there exists some i such that

where cl denotes the closure operator.

Theorem. A family of continuous maps {fi: XYi} separates points from closed sets if and only if the cylinder sets , for U open in Yi, form a base for the topology on X.

It follows that whenever {fi} separates points from closed sets, the space X has the initial topology induced by the maps {fi}. The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

If the space X is a T0 space, then any collection of maps {fi} that separates points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.

Categorical description

In the language of category theory, the initial topology construction can be described as follows. Let be the functor from a discrete category to the category of topological spaces which maps . Let be the usual forgetful functor from to . The maps can then be thought of as a cone from to . That is, is an object of the category of cones to . More precisely, this cone defines a -structured cosink in .

The forgetful functor induces a functor . The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from to , i.e.: a terminal object in the category .
Explicitly, this consists of an object in together with a morphism such that for any object in and morphism there exists a unique morphism such that the following diagram commutes:


The assignment placing the initial topology on extends to a functor which is right adjoint to the forgetful functor . In fact, is a right-inverse to ; since is the identity functor on .

See also

Related Research Articles

In mathematics, specifically category theory, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms.

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets.

General topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.

In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.

In mathematics, the smash product of two pointed spaces and is the quotient of the product space X × Y under the identifications (xy0) ∼ (x0y) for all x ∈ X and y ∈ Y. The smash product is itself a pointed space, with basepoint being the equivalence class of. The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints.

Cone (topology) in topology

In topology, especially algebraic topology, the coneCXof a topological spaceX is the quotient space:

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x0, that remains unchanged during subsequent discussion, and is kept track of during all operations.

In general topology and related areas of mathematics, the final topology on a set , with respect to a family of functions into , is the finest topology on that makes those functions continuous.

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an exponential law.

In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.

In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.