category of topological spaces oor Spaans
category of topological spaces
Vertalings in die woordeboek Engels - Spaans
Categoría de espacios topológicos
en
large category whose objects are topological spaces and whose morphisms are continuous maps
Geskatte vertalings
Vertoon algoritmies gegenereerde vertalings
voorbeelde
Advanced filtering
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For instance, in computer science and physics, there are categories of topological spaces and other complex structures arising.
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In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms.
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The setting for these was the 'category of topological spaces and continuous maps.'
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These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions.
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These are precisely the finitely generated members of the category of topological spaces and continuous maps.
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Here the quotient is in the category of topological spaces, rather than groups or rings.
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Thus the category of completely regular spaces CReg is a reflective subcategory of Top, the category of topological spaces.
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In other words, we have a functor from the category of topological spaces with base point to the category of groups.
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Direct limits in the category of topological spaces are given by placing the final topology on the underlying set-theoretic direct limit.
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The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification.
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Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
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Similarly, in the category of pointed topological spaces, every singleton is a zero object.
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Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points.
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Further insights result when categories of orders are found categorically equivalent to other categories, for example of topological spaces.
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Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober spaces and spatial locales, are to be found in the article on Stone duality.
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Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
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In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum.
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If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category.
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Category of topological spaces
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This shows that the product space is a product in the category of topological spaces.
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If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels.
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Similarly, the empty space is the unique initial object in the category of topological spaces; every one-point space is a terminal object in this category.
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If X is a topological space, we can consider the abelian category of all sheaves of abelian groups on X. The functor which associates to each sheaf F the group of global sections F(X) is left-exact.
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Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.
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Every continuous map f : X → Y between topological spaces induces a functor f ∗ from the category of sheaves (of sets, or abelian groups, or rings...) on X to the corresponding category of sheaves on Y, the direct image functor.
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