connectedness

connectedness
See connectedly.

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      in mathematics, fundamental topological property of sets that corresponds with the usual intuitive idea of having no breaks. It is of fundamental importance because it is one of the few properties of geometric figures that remains unchanged after a homeomorphism—that is, a transformation in which the figure is deformed without tearing or folding. A point is called a limit point of a set in the Euclidean plane if there is no minimum distance from that point to members of the set; for example, the set of all numbers less than 1 has 1 as a limit point. A set is not connected if it can be divided into two parts such that a point of one part is never a limit point of the other part. The set is connected if it cannot be so divided. For example, if a point is removed from an arc, any remaining points on either side of the break will not be limit points of the other side, so the resulting set is disconnected. If a single point is removed from a simple closed curve such as a circle or polygon, on the other hand, it remains connected; if any two points are removed, it becomes disconnected. A figure-eight curve does not have this property because one point can be removed from each loop and the figure will remain connected. Whether or not a set remains connected after some of its points are removed is one of the principal ways of classifying figures in topology.

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Universalium. 2010.

Look at other dictionaries:

  • connectedness — n. 1. the state of being connected. Syn: connection. [WordNet 1.5] 2. a relation between things or events (as in the case of one causing the other or sharing features with it). Syn: connection, connexion. [WordNet 1.5] …   The Collaborative International Dictionary of English

  • connectedness — UK US /kəˈnektɪdnəs/ noun [U] ► the state of being connected and having a close relationship with other things or people: »The book s theme is the conflict between connectedness, a product of the revolution in communications and technology, and… …   Financial and business terms

  • connectedness — index adherence (adhesion), association (connection), chain (nexus), coherence, continuity …   Law dictionary

  • Connectedness — This article is about mathematics. For other uses, see Connectedness (disambiguation). In mathematics, connectedness is used to refer to various properties meaning, in some sense, all one piece . When a mathematical object has such a property, we …   Wikipedia

  • connectedness — con|nect|ed|ness [kəˈnektıdnıs] n [U] 1.) the feeling people have that they are members of a group in society and that they share particular qualities with other members of that group ▪ Human beings have a need for both independence and… …   Dictionary of contemporary English

  • connectedness — noun see connected …   New Collegiate Dictionary

  • connectedness — noun The state or quality of being connected …   Wiktionary

  • connectedness — n. state of being attached, ability to be connected …   English contemporary dictionary

  • connectedness — con·nect·ed·ness …   English syllables

  • connectedness — noun 1. the state of being connected the connection between church and state is inescapable • Syn: ↑connection, ↑link • Ant: ↑disconnectedness • Derivationally related forms: ↑connected, ↑ …   Useful english dictionary

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