# Equivalence of topological definitions of connectedness

• We know a set is connected if it cannot be written as a union of two disjoint non-empty open sets.
• We know a set is connected if it cannot be written as a union of two disjoint non-empty closed sets.

How can these definitions be equivalent? By the first definition, the set $$[1,2] \cup [3,4]$$ is connected (since it can't be written as the union of two disjoint non-empty open sets). But it isn't connected by the second definition. Of course, the set is clearly disconnected, so I think I'm misunderstanding something about the first definition, but I'm not sure what...

• Here, open refers to the subspace topology. The sets $[1,2]$ and $[3,4]$ are both open and closed as subsets of the topological space $[1,2]\cup[3,4]$. Commented Apr 3, 2021 at 23:05
• In topology, always distinguish between a subset and a subspace. Even though we sometimes speak of one when we mean the other. This is called abuse of terminology, which is common and often tolerated, along with abuse of notation. It drives students crazy. Commented Apr 4, 2021 at 2:46

Adding the answer by Vercassivelaunos here (as a community wiki) so I can mark the question answered:

Here, open refers to the subspace topology. The sets [1,2] and [3,4] are both open and closed as subsets of the topological space [1,2]∪[3,4].

Thank you very much for your help!