Definition:Irreducible Space/Linguistic Note

Linguistic Note on Irreducible Space

The thinking behind applying the word irreducible to the concept of an irreducible space arises as follows.

By definition, we cannot express $X$ as the union of two proper closed sets of $T$.

Expressing a space as the union of two smaller closed sets can be considered as reducing it.

There are parallels with the concept of an irreducible element of a ring in abstract algebra, which cannot be written as a product of two non-units.

The terminology comes from the Zariski topology in the context of algebraic geometry, where there is a direct link to irreducible varieties.

While the name hyperconnected space is more intuitively clear, and bears a pleasing antithesis with the concept of ultraconnected space, it is considered old-fashioned.

Sources

• Henno Brandsma (https://math.stackexchange.com/users/4280/henno-brandsma), What is the reason for an "irreducible" (topological) space to be so called?, URL (version: 2020-05-22): https://math.stackexchange.com/q/3686211