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: 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. Every subset is dense or nowhere dense in X. No two points can be separated by disjoint neighbourhoods.
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.
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 .
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 schemes , are 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 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 (by the genus–degree formula)
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.