Hyperconnected space

In the mathematical field of topology, a hyperconnected space or irreducible 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.
Two examples of hyperconnected spaces from point set topology are the cofinite topology on any infinite set and the right order topology on formula_1.
In algebraic geometry, taking the spectrum of a ring whose reduced ring is an integral domain is an irreducible topological space—applying the lattice theorem to the nilradical, which is within every prime, to show the spectrum of the quotient map is a homeomorphism, this reduces to the irreducibility of the spectrum of an integral domain. For example, the schemesformula_2 , formula_3are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial factorization). A non-example is given by the normal crossing divisorformula_4since the underlying space is the union of the affine planes formula_5, formula_6, and formula_7. Another non-example is given by the schemeformula_8where formula_9 is an irreducible degree 4 homogeneous polynomial. This is the union of the two genus 3 curves (by the genus–degree formula)formula_10
Hyperconnectedness vs. connectedness.
Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-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 (non-disjoint) closed sets.
Irreducible components.
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.
Every irreducible subset of a space "X" is contained in a (not necessarily unique) irreducible component of "X". In particular, every point of "X" is contained in some irreducible component of "X". 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.
The irreducible components of a Hausdorff space are just the singleton sets.
Since every irreducible space is connected, the irreducible components will always lie in the connected components.
Every Noetherian topological space has finitely many irreducible components.