Hyperconnected space

Definition

Let $X$ be a topological space. The space $X$ is hyperconnected, or irreducible, if the following two conditions hold.

$X$ is nonempty.
For all nonempty open sets $U, V \subseteq X$ , the intersection $U \cap V$ is nonempty.

Remarks

Equivalent forms: every nonempty open subset of $X$ is dense in $X$ . Also equivalent: $X$ is nonempty and is not the union of two closed subsets different from $X$ .

Example: an infinite set $X$ with the cofinite topology. Nonempty open sets have finite complements, so the union of the complements of $U$ and $V$ is finite and cannot be all of $X$ .

Non-example: a Hausdorff space with at least two points fails clause 2. Two distinct points have disjoint open neighbourhoods.

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Open set concept
Topological space concept
Hyperconnected space definition

Tag	`5HDNJWZ6K9`
URI	`m2://5HDNJWZ6K9`
Kind	definition
Writing	W0 stub
MSC	54D05, 14A05
Defines	Hyperconnected space
Uses	Topological space, Open set
Changed	2 November 2026

Hyperconnected space

Definition

Remarks

Reading path

Further reading

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