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In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two non-empty 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:
- 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).
Hyperconnectedness vs. connectedness
Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to the definition of connectedness, in which the open sets are disjoint.
For example, the space of reals with the standard topology is connected but not hyperconnected. This is because it cannot be written as a union of two disjoint open sets, but it can be written as a union of two (non-disjoint) closed sets.
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.
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.
An irreducible component in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed.
Unlike the connected components of a space, the irreducible components need not be disjoint (i.e. they need not form a partition). In general, the irreducible components will overlap. Since every irreducible space is connected, the irreducible components will always lie in the connected components.
The irreducible components of a Hausdorff space are just the singleton sets.
Every subset of a Noetherian topological space is Noetherian, and hence has finitely many irreducible components.