Irreducible set

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.

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).

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

${\displaystyle {\text{Spec}}\left({\frac {\mathbb {Z} [x,y,z]}{x^{4}+y^{3}+z^{2}}}\right)}$ , ${\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(y^{2}z-x(x-z)(x-2z))}}\right)}$

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