# Error? An open subset of $\mathbb R^p$ is connected if and only if it can be expressed as the union of two disjoint non-empty open sets.

I believe the book which I am reading has a printing error. One of the lemmas reads like this

An open subset of $\mathbb R^p$ is connected $\iff$if it can be expressed as the union of two disjoint non-empty open sets.

I think, instead, the statement should be : An open subset of $\mathbb R^p$ is disconnected $\iff$if it can be expressed as the union of two disjoint non-empty open sets.

Am I correct?

• Yes. Alternatively, there is a "not" missing. – Daniel Fischer Sep 1 '14 at 12:44
• That sounds correct but is not really, consider $p=1$ and $[0,1] \setminus \{\frac12\}$. This cannot be expressed as the union of two non-empty disjoint open sets. I think it needs to say can be covered by two disjoint open sets – AlexR Sep 1 '14 at 12:45
• @DanielFischer What about my counterexample? – AlexR Sep 1 '14 at 12:45
• @AlexR "An open subset of $\mathbb{R}^p$"; otherwise, one would consider relatively open sets. – Daniel Fischer Sep 1 '14 at 12:47