In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Jump to: navigation, search In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ...

Definition

Given a set X and a family of topological spaces Y_i with functions In mathematics, an index set is another name for a function domain. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

the initial topology τ on X is the coarsest topology such that each Jump to: navigation, search In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. ...

is continuous. In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ...

Explicitly, the initial topology may be described as the topology generated by sets of the form , where U is an open set in Y_i. In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

Examples

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

• The subspace topology is the initial topology on the subspace with respect to the inclusion map.
• The product topology is the initial topology with respect to the family of projection maps.
• The inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
• The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space.
• Given a family of topologies {τ_i} on a fixed set X the initial topology on X with respect to the functions id_X : X → (X, τ_i) is the supremum (or join) of the topologies {τ_i} in the lattice of topologies on X. That is, the initial topology τ is the topology generated by the union of the topologies {τ_i}.

This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, inclusion is a partial order on sets. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... Jump to: navigation, search In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. X × Y = { (x, y... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In mathematics, weak topology is an alternative term for initial topology. ... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... Jump to: navigation, search In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, an index set is another name for a function domain. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... Jump to: navigation, search In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Properties

The initial topology on X can be characterized by the following universal property: a function g from some space Z to X is continuous if and only if is continuous for each i ∈ I. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

By the universal property of the product topology we know that any family of continuous maps f_i : X → Y_i determines a unique continuous map Jump to: navigation, search Image File history File links InitialTopology-01. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

If the family of maps {f_i} separates points in X (i.e. for all x ≠ y in X there exists some f_i such that f_i(x) ≠ f_i(y)) then the map f will be a topological embedding if and only if X has the initial topology determined by the maps f_i. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

In the language of category theory, the initial topology construction can be described as follows. Let Y be a functor from a discrete category J to the category of topological spaces Top which selects the spaces Y_i for i in J. Let Δ 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 (Δ ↓ Y) is then the category of all cones to Y, i.e. objects in (Δ ↓ Y) are pairs (X, f) where f_i : X → Y_i is a family of continuous maps on X. If U is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set^J then the comma category (Δ′ ↓ UY) is the category of all cones to UY. The initial topology construction can then be described as a functor from (Δ′ ↓ UY) to (Δ ↓ Y). This functor is right adjoint to the corresponding forgetful functor. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In category theory, a discrete category is a category whose only morphisms are the identity morphisms. ... The category Top has topological spaces as objects and continuous maps as morphisms. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... A comma category is a construction in category theory, a branch of mathematics. ... A forgetful functor is a type of functor in mathematics. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

See also

• Final Topology