Hyperconnected Space

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

For a topological space X the following conditions are equivalent:

• no two nonempty open sets are disjoint
• X cannot be written as the union of two proper closed sets
• every nonempty open set is dense in X
• the interior of every proper closed set is empty

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).

The (nonempty) open subsets of a hyperconnected space are "large" in the sense that each one is dense in X and any pair of them intersects. Thus, a hyperconnected space cannot be Hausdorff unless it contains only a single point.

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

Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected). The continuous image of a hyperconnected space is hyperconnected. In particular, any continuous function from a hyperconnected space to a Hausdorff space must be constant. It follows that every hyperconnected space is pseudocompact.

Every open subspace of a hyperconnected space is hyperconnected. A closed subspace need not be hyperconnected, however, the closure of any hyperconnected subspace is always hyperconnected.