---
tag: 38CDCK1RYB
kind: concept
name: Hyperconnected space
msc: [54D05, 14A05]
specializes: [0V032VWTHD]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperconnected_space
same_as:
  wikidata: Q1673182
  wikipedia: Hyperconnected space
---

## Summary

A hyperconnected space, also called an irreducible space, is a nonempty [[topological space|0CQHFPGJ45]] in which any two nonempty open sets meet.
Equivalently, it is not the union of two proper closed subsets.
Every hyperconnected space is [[connected|0V032VWTHD]], but the condition is much stronger.
Irreducible spaces are basic in algebraic geometry, where irreducible varieties and the spectra of integral domains have this property.

## Explanations

### L1 human

In a hyperconnected space, every nonempty open set is spread all over the space: it is dense.
This never happens in a Hausdorff space with two or more points, where two points have disjoint neighbourhoods.
An infinite set with the [[cofinite topology|0B906ZBTMZ]] is hyperconnected, since two nonempty open sets miss only finitely many points each.
Such coarse topologies appear naturally as Zariski topologies in algebraic geometry.