general topology and related areas of mathematics, the initial topology (or induced topology or strong topology or limit topology or projective topology) on...
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In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators,...
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natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which...
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In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can...
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general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It...
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is the initial topology, which for a given family of functions from a set X {\displaystyle X} into topological spaces is the coarsest topology on X {\displaystyle...
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coarsest topology may refer to: Initial topology, the most coarse topology in a certain category of topologies Trivial topology, the most coarse topology possible...
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the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly...
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Continuous function (redirect from Continuous function (topology))
an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus...
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Category of topological spaces (redirect from Categorical topology)
a unique initial lift ( A → A i ) I {\displaystyle (A\to A_{i})_{I}} . In Top the initial lift is obtained by placing the initial topology on the source...
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Subbase (redirect from Subbase (topology))
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X {\displaystyle X} with topology τ {\displaystyle \tau } is a subcollection...
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Tychonoff space (category Topology)
well-behaved with respect to initial topologies. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property...
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Sierpiński space (redirect from Sierpinski topology)
generalized by the notion of a presheaf. Any topological space X has the initial topology induced by the family C ( X , S ) {\displaystyle C(X,S)} of continuous...
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Inverse limit (redirect from Limit topology)
constructed by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology. The set of infinite strings...
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areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric...
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Vague topology Warsaw circle List of natural topologies. Adjunction space Disjoint union (topology) Extension topology Initial topology Final topology Product...
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topology. The simplest cases (which nevertheless cover many examples) are the initial topology and the final topology (Willard (1970)). The initial topology...
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Topological indistinguishability (category General topology)
In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are...
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Locally convex topological vector space (redirect from Locally convex topology)
weakest topology making all linear functionals in F {\displaystyle F} continuous. This is known as the weak topology or the initial topology determined...
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Discrete space (redirect from Discrete topology)
certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular...
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0. The uniform topology generated by the above uniformity is the initial topology induced by the family C(X). In general, this topology will be coarser...
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analogue to final topologies) exist. But there is a catch: While the initial structure of the above property is in fact the usual initial topology on V {\displaystyle...
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into X . {\displaystyle X.} Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there...
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Lately, topology control algorithms have been divided into two subproblems: topology construction, in charge of the initial reduction, and topology maintenance...
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= [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. A loop may also be seen as a continuous...
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among topologies, applying speciation (the evolution of species) to preserve innovations, and developing topologies incrementally from simple initial structures...
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Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational...
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use in routing of mesh networks, specifically pertaining to their initial topology. It was developed in 1991 by Aaron Kershenbaum, Parviz Kermani, and...
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triangulations established a new branch in topology, namely the piecewise-linear-topology (short PL-topology). Its main purpose is topological properties...
33 KB (5,219 words) - 17:37, 29 September 2024
In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent...
10 KB (1,728 words) - 01:51, 19 January 2024