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 subsets (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". Due to the condition about neighborhoods of distinct points being in a sense the opposite of the Hausdorff property, some authors call such spaces anti-Hausdorff.
The empty set is vacuously a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets). However some authors, especially those interested in applications to algebraic geometry, add an explicit condition that an irreducible space must be nonempty. 
An irreducible set is a subset of a topological space for which the subspace topology is irreducible.
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.
 Proof: "Let formula_11 be an open subset. Any two disjoint open subsets of formula_12 would themselves be disjoint open subsets of formula_13. So at least one of them must be empty."
 Proof: "Suppose formula_14 is a dense subset of formula_13 and formula_16 with formula_17, formula_18 closed in formula_14. Then formula_20. Since formula_13 is hyperconnected, one of the two closures is the whole space formula_13, say formula_23. This implies that formula_17 is dense in formula_14, and since it is closed in formula_14, it must be equal to formula_14."
 Counterexample: "formula_28 with formula_29 an algebraically closed field (thus infinite) is hyperconnected in the Zariski topology, while formula_30 is closed and not hyperconnected."
 Proof: "Suppose formula_31 where formula_14 is irreducible and write formula_33 for two closed subsets formula_34 (and thus in formula_13). formula_36 are closed in formula_14 and formula_38 which implies formula_39 or formula_40, but then formula_41 or formula_42 by definition of closure."
 Proof: "Firstly, we notice that if formula_47 is a non-empty open set in formula_13 then it intersects both formula_49 and formula_50; indeed, suppose formula_51, then formula_52 is dense in formula_49, thus formula_54 and formula_55 is a point of closure of formula_52 which implies formula_57 and a fortiori formula_58. Now formula_59 and taking the closure formula_60 therefore formula_47 is a non-empty open and dense subset of formula_13. Since this is true for every non-empty open subset, formula_13 is irreducible."
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.
Notes.
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