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- Open_set abstract "In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain an open ball around each of their points (or, equivalently, a set is open if it doesn't contain any of its boundary points); however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open (called the discrete topology), or no set can be open but the space itself (the indiscrete topology).In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets.Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mathematics. Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.".
- Open_set thumbnail Red_blue_circle.svg?width=300.
- Open_set wikiPageID "39358".
- Open_set wikiPageRevisionID "598236968".
- Open_set hasPhotoCollection Open_set.
- Open_set id "2925".
- Open_set id "p/o068310".
- Open_set title "Open Set".
- Open_set title "Open set".
- Open_set subject Category:General_topology.
- Open_set comment "In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.".
- Open_set label "Conjunto aberto".
- Open_set label "Conjunto abierto".
- Open_set label "Insieme aperto".
- Open_set label "Offene Menge".
- Open_set label "Open set".
- Open_set label "Open verzameling".
- Open_set label "Ouvert (topologie)".
- Open_set label "Zbiór otwarty".
- Open_set label "Открытое множество".
- Open_set label "مجموعة مفتوحة".
- Open_set label "开集".
- Open_set label "開集合".
- Open_set sameAs Otevřená_množina.
- Open_set sameAs Offene_Menge.
- Open_set sameAs Conjunto_abierto.
- Open_set sameAs Ouvert_(topologie).
- Open_set sameAs Himpunan_terbuka.
- Open_set sameAs Insieme_aperto.
- Open_set sameAs 開集合.
- Open_set sameAs 열린_집합.
- Open_set sameAs Open_verzameling.
- Open_set sameAs Zbiór_otwarty.
- Open_set sameAs Conjunto_aberto.
- Open_set sameAs m.09syn.
- Open_set sameAs Q213363.
- Open_set sameAs Q213363.
- Open_set wasDerivedFrom Open_set?oldid=598236968.
- Open_set depiction Red_blue_circle.svg.
- Open_set isPrimaryTopicOf Open_set.