Space with finitely many connected components

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Definition

A space with finitely many connected components is a topological space satisfying the following equivalent conditions:

  1. It has finitely many connected components.
  2. It can be expressed as a disjoint union of finitely many pairwise disjoint clopen subsets each of which is a connected space with the subspace topology.
  3. It has finitely many clopen subsets (the clopen subsets will precisely be all possible unions of the connected components).

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
connected space |FULL LIST, MORE INFO
space with finitely many irreducible components |FULL LIST, MORE INFO
irreducible space Connected space|FULL LIST, MORE INFO
Noetherian space |FULL LIST, MORE INFO
space with finitely many path components |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
space with finitely many quasicomponents |FULL LIST, MORE INFO
space in which all connected components are open |FULL LIST, MORE INFO
space in which the connected components concide with the quasicomponents Space in which all connected components are open|FULL LIST, MORE INFO