Space in which all connected components are open

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Definition

A space in which all connected components are open is a topological space satisfying the following equivalent conditions:

  1. Every point is contained in an open subset that is connected in the subspace topology.
  2. Every connected component is an open subset.
  3. Every connected component is a clopen subset.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
connected space
locally connected space
space with finitely many connected components

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
space in which the connected components coincide with the quasicomponents