In mathematics, a hyperconnected 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.
Contents
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
An example of a hyperconnected space from point set topology is the cofinite topology on any infinite space.
In algebraic geometry, taking the spectrum of a ring whose reduced ring is an integral domain is an irreducible topological space. 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 divisor
since the underlying space is the union of the affine planes
where
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 reals 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.
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 (though not necessarily path-connected or locally path-connected).
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. A closed subspace need not be hyperconnected, however, the closure of any hyperconnected subspace is always hyperconnected.
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.
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. 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.