# hyperconnected space

A topological space $X$ 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 sets). Hyperconnected spaces are sometimes known as irreducible sets (http://planetmath.org/IrreducibleClosedSet).

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

Any infinite set with the cofinite topology 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 topology, are also hyperconnected.

Title hyperconnected space HyperconnectedSpace 2013-03-22 14:20:30 2013-03-22 14:20:30 yark (2760) yark (2760) 10 yark (2760) Definition msc 54D05 hyper-connected space UltraconnectedSpace IrreducibleClosedSet hyperconnected hyper-connected