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# hyperconnected space

A topological space $X$ is said to be *hyperconnected* if no pair of nonempty open sets of $X$ is disjoint (or, equivalently, if $X$ is not the union of two proper closed sets).
Hyperconnected spaces are sometimes known as irreducible sets.

All hyperconnected spaces are connected, locally connected, and pseudocompact.

Any infinite set with the cofinite topology is an example of a hyperconnected space. Similarly, any uncountable set with the cocountable topology is hyperconnected. Affine spaces and projectives spaces over an infinite field, when endowed with the Zariski topology, are also hyperconnected.

Defines:

hyperconnected, hyper-connected

Related:

UltraconnectedSpace, IrreducibleClosedSet

Synonym:

hyper-connected space

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54D05*no label found*

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