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.