hyperconnected space

A topological spaceMathworldPlanetmath X is said to be hyperconnected if no pair of nonempty open sets of X is disjoint (or, equivalently, if X is not the union of two proper closed setsPlanetmathPlanetmath). Hyperconnected spaces are sometimes known as irreducible sets (http://planetmath.org/IrreducibleClosedSet).

All hyperconnected spaces are connectedPlanetmathPlanetmath, locally connected, and pseudocompact.

Any infinite setMathworldPlanetmath with the cofinite topologyMathworldPlanetmath is an example of a hyperconnected space. Similarly, any uncountable set with the cocountable topology is hyperconnected. Affine spaces and projectives spaces over an infinite field, when endowed with the Zariski topologyMathworldPlanetmath, are also hyperconnected.

Title hyperconnected space
Canonical name HyperconnectedSpace
Date of creation 2013-03-22 14:20:30
Last modified on 2013-03-22 14:20:30
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Definition
Classification msc 54D05
Synonym hyper-connected space
Related topic UltraconnectedSpace
Related topic IrreducibleClosedSet
Defines hyperconnected
Defines hyper-connected