Connected space and his relathionship with his own subset

Question

I saw this proposition without the prove and I thought that it was wrong.

$X$ connected if, and only if, any own subset $A$ from $X$ has some point of his border.In other words, his border isn´t empty.

I thought it can´t be true because if I take $B(0;1)$ that is connected I can take a own subset $B(0;1/2)$ that is open and doesn´t have points of his border.

I don´t know if I am misunderstanding something.

Thanks in advance for your help!

Answer 1

The correct statement is the following: $X$ is connected iff the boundary of any non-empty proper subset is non-empty. Proof: $X$ is connected iff the only open and closed subsets are empty set and $X$. A subset $A$ of $X$ has empty boundary iff $\overline {A}=A^{0}$ (i.e. closure of $A$ equals its interior) and the condition $\overline {A}=A^{0}$ implies $A$ is both open and closed. (Note that $\overline {A}=\emptyset$ implies $A=\emptyset$ and $A^{0}=X$ implies that $A=X$).