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29 relations: Adjoint functors, Category of topological spaces, Category theory, Coherent topology, Comma category, Comparison of topologies, Cone (category theory), Continuous function, Diagonal functor, Direct limit, Discrete category, Disjoint union (topology), Family of sets, Forgetful functor, Functor, Functor category, General topology, If and only if, Inclusion map, Induced topology, Infimum and supremum, Initial topology, Intersection (set theory), Mathematics, Quotient space (topology), Set (mathematics), Sheaf (mathematics), Topological space, Universal property.
Adjoint functors
In mathematics, specifically category theory, adjunction is a possible relationship between two functors.
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Category of topological spaces
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.
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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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Coherent topology
In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces.
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Comma category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory.
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Comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.
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Cone (category theory)
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Diagonal functor
In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a).
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Direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.
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Discrete category
In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
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Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology.
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Family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.
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Forgetful functor
In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output.
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Functor
In mathematics, a functor is a map between categories.
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Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors.
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General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element, x, of A to x, treated as an element of B: A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (On the other hand, this notation is sometimes reserved for embeddings.) This and other analogous injective functions from substructures are sometimes called natural injections.
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Induced topology
In topology and related areas of mathematics, an induced topology on a topological space is a topology which makes the inducing function continuous from/to this topological space.
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Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
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Initial topology
In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that makes those functions continuous.
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Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Quotient space (topology)
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.
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Redirects here:
Direct limit topology, Final limit topology, Inductive limit topology, Inductive topology, Strongest topology.
References
[1] https://en.wikipedia.org/wiki/Final_topology
