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Closure operator

In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)} from the power set of S to itself that satisfies the following conditions for all sets X , Y ⊆ S {\displaystyle X,Y\subseteq S}

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Closure operators on partially ordered sets

Binary relationAntisymmetricPointwise orderIdentity functionDual
19 related topics

Examples

ClosureTopologyLinear spanVector spaceConvex hull
16 related topics

Closure operators in algebra

Universal algebraAlgebraSubalgebraGroupSubgroup
16 related topics

History

E. H. MooreFrigyes RieszErnst SchröderRichard DedekindGeorg Cantor
5 related topics

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History

Closed sets

Examples

Closure operators in topology

Closure operators in algebra

Closure operators in logic

Closure operators on partially ordered sets

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Closure operator

Nodes82
Edges81
Triples133
Avg. degree1.98
Density0.02439
Components1

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Closure operator

Top relations

related to References · 81
Closure operator → Abstract Logics, Alfred, Alfred Tarski, Algebra Universalis, American Mathematical Society, Annals, Applications, Approximate Reasoning, Available, Bernhard, Birkhaeuser, Blyth, Boston MA, Brown, Burris, Cambridge University Press, Castellini, Categorical, Conceptual Exploration, Continuous Lattices
related to Closure operators in logic · 14
Closure operator → Also, Brown, Call, Consider, For, Gerla, In, Lloyd, More, Suppose, Suszko, Tarski, Then, This
related to Consequence operators · 7
Closure operator → Alfred Tarski, Around, In, Mathematically, Nowadays, Tarski, The
related to history · 7
Closure operator → Ernst Schröder, Frigyes Riesz, Georg Cantor, Introduction, Moore, Richard Dedekind, Though
see also · 6
Closure operator → Algebraic, All, Axioms, Closure, Largest, Particular
related to Closed sets · 5
Closure operator → Any, Conversely, In, The, There
related to Closure operators in algebra · 5
Closure operator → Every, Finitary, Perhaps, Similarly, This
related to Pseudo-closed sets · 3
Closure operator → Each, Formally, These
related to Closure operators in topology · 2
Closure operator → Conversely, The
related to Examples · 2
Closure operator → Other, The

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Important terminology

closure operator set operators sets closed subset cl every logic finitary function called ordered algebraic algebra space partially topology given

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Closure operatorrelated to Closed setsThe0.60section
Closure operatorrelated to Closed setsAny0.60section
Closure operatorrelated to Closed setsIn0.60section
Closure operatorrelated to Closed setsConversely0.60section
Closure operatorrelated to Closed setsThere0.60section
Closure operatorrelated to Closure operators in algebraFinitary0.60section
Closure operatorrelated to Closure operators in algebraEvery0.60section
Closure operatorrelated to Closure operators in algebraThis0.60section
Closure operatorrelated to Closure operators in algebraPerhaps0.60section
Closure operatorrelated to Closure operators in algebraSimilarly0.60section
Closure operatorrelated to Closure operators in logicSuppose0.60section
Closure operatorrelated to Closure operators in logicConsider0.60section

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