In mathematics, a hyperconnected space is a topological spaceX 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:
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.
are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials. A non-example is given by the normal crossing divisorsince the underlying space is the union of the affine planes,, and. Another non-example is given by the schemewhere is an irreducible degree 4 homogeneous polynomial. This is the union of the two genus 3 curves
Hyperconnectedness vs. connectedness
Every hyperconnected space is both connected and locally connected. 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 real numbers 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 closed sets.
Properties
The 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.
Every hyperconnected space is both connected and locally connected.
Since the closure of every non-empty open set in a hyperconnected space is the whole space, which is an open set, every hyperconnected space is extremally disconnected.
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. Proof. Let be an open subset. Any two disjoint open subsets of would themselves be disjoint open subsets of. So at least one of them must be empty.
A closed subspace need not be hyperconnected. Counterexample. with an algebraically closed field is hyperconnected in the Zariski topology, while is closed and not hyperconnected.
The closure of any hyperconnected subspace is always hyperconnected. Proof. Suppose where is irreducible and write for two closed subsets . are closed in and which implies or, but then or by definition of closure.
A space which can be written as with open and irreducible such that is irreducible. Proof. Firstly, we notice that if is a non-empty open set in then it intersects both and ; indeed, suppose, then is dense in, thus and is a point of closure of which implies and a fortiori. Now and taking the closure therefore is a non-empty open and dense subset of. Since this is true for every non-empty open subset, is irreducible.
Irreducible components
An irreducible component in a topological space is a maximal irreducible subset. The irreducible components are always closed. Unlike the connected components of a space, the irreducible components need not be disjoint. 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.
