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 subsets. 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 subsets.
Every nonempty open set is dense in X.
Every open set is connected.
The interior of every proper closed subset of X 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.
The empty set is vacuously a hyperconnected or irreducible space under the definition above. 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.
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. A non-example is given by the normal crossing divisor

since the underlying space is the union of the affine planes,, and. Another non-example is given by the scheme

where 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 need not 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 the union of two disjoint open sets, but it can be written as the union of two closed sets.

Properties

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.
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 image of a hyperconnected space under a continuous function 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.
More generally, every dense subset of a hyperconnected space is hyperconnected.
A closed subspace of a hyperconnected space need not be hyperconnected.
The closure of any irreducible set is irreducible.
A space which can be written as with open and irreducible such that is irreducible.

Irreducible components

An irreducible component in a topological space is a maximal irreducible subset. The irreducible components are always closed.
Every irreducible subset of a space X is contained in a 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. 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.