In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
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| - Kuratowského axiomy uzávěru (cs)
- Assiomi di chiusura di Kuratowski (it)
- Kuratowski closure axioms (en)
- 庫拉托夫斯基閉包公理 (zh)
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| - Kuratowského axiomy uzávěru je sada axiomů v topologii a příbuzných oblastech matematiky, které lze použít pro definici topologického prostoru na množině. Jsou ekvivalentní s častěji používanou definicí otevřené množiny. Axiomy formalizoval Kazimierz Kuratowski, a myšlenku dále rozvinuli další matematici, mimo jiné Wacław Sierpiński a . Pro definici topologické struktury lze použít i podobnou množinu axiomů, která používá duální pojem operátoru vnitřku množiny. (cs)
- In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator. (en)
- In topologia e nella branche matematiche ad essa collegate gli assiomi di chiusura di Kuratowski sono un gruppo di assiomi che possono essere utilizzati per definire una struttura topologica su un insieme. Sono equivalenti alla più comune definizione basata sugli insiemi aperti. Furono introdotti per la prima volta da Kazimierz Kuratowski, in una forma lievemente differente applicabile esclusivamente agli spazi di Hausdorff. Un gruppo simile di assiomi può essere utilizzato per definire una struttura topologica sfruttando esclusivamente la nozione duale di operatore interno. (it)
- 庫拉托夫斯基閉包公理可來定義一個集上的拓扑結構，它和以開集作定義拓樸結構的公理等價。 (zh)
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| - Proof 1. (en)
- Proof 2. (en)
- Proof 3. (en)
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| - Kuratowského axiomy uzávěru je sada axiomů v topologii a příbuzných oblastech matematiky, které lze použít pro definici topologického prostoru na množině. Jsou ekvivalentní s častěji používanou definicí otevřené množiny. Axiomy formalizoval Kazimierz Kuratowski, a myšlenku dále rozvinuli další matematici, mimo jiné Wacław Sierpiński a . Pro definici topologické struktury lze použít i podobnou množinu axiomů, která používá duální pojem operátoru vnitřku množiny. (cs)
- In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator. (en)
- In topologia e nella branche matematiche ad essa collegate gli assiomi di chiusura di Kuratowski sono un gruppo di assiomi che possono essere utilizzati per definire una struttura topologica su un insieme. Sono equivalenti alla più comune definizione basata sugli insiemi aperti. Furono introdotti per la prima volta da Kazimierz Kuratowski, in una forma lievemente differente applicabile esclusivamente agli spazi di Hausdorff. Un gruppo simile di assiomi può essere utilizzato per definire una struttura topologica sfruttando esclusivamente la nozione duale di operatore interno. (it)
- 庫拉托夫斯基閉包公理可來定義一個集上的拓扑結構，它和以開集作定義拓樸結構的公理等價。 (zh)
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