Hyperconnected space

In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name irreducible space is preferred in algebraic geometry.

For a topological space X the following conditions are equivalent:

A space which satisfies any one of these conditions is called hyperconnected or irreducible.

An irreducible set is a subset of a topological space for which the subspace topology is irreducible. Some authors do not consider the empty set to be irreducible (even though it vacuously satisfies the above conditions).

Examples

Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology on an algebraic variety.

Hyperconnectedness vs. connectedness

Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected).