If every non-trivial subset of a set C has non-empty boundary in C, then C is connected

Question

I came across this proposition in Functional analysis book by Joseph Muscat. I am having problem in applying this to following example. Let's say C = [-1,1] \ {0}, then C is not connected. But I am not able to find a subset with non-empty boundary. Subset A = [-1,0) has non-empty boundary {-1} in C. Please help me understand what I am missing. Thanks.

Answer 1

The boundary of your set $A$ is empty. $-1$ is an interior point of $A$ in your space $C$ since $[-1,0)$ is an open subset of $C$ containing $-1$ which is contained in $A$. .