Hyperconnected space
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In mathematics, a hyperconnected space (or irreducible space) is a topological space X that satisfies any of the following equivalent conditions:
- 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
The (nonempty) open sets of a hyperconnected space are "large" in the sense that they are dense in X and every pair of them intersects. Such a 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. The closure of any hyperconnected subspace is hyperconnected.
