Help me please. I can not prove the following.
Prove that a point belongs to the lock of $\displaystyle A$ if and only if is a interior point or a frontier point of $\displaystyle A$.
Thanks
Some translations are in order here.
‘lock’ must mean closure; ‘frontier’ must mean boundary.
A point is in the closure of the set iff every open set containing the point contains a point of the set.
Thus, if we have a interior point or a frontier point of $\displaystyle A$ then by definition it is in the closure.
Suppose $\displaystyle p\in \overline{A}$, the closure.
If $\displaystyle p$ is an interior point of $\displaystyle A$ we are done.
So what does it mean to say that $\displaystyle p$ is not an interior point?
Now you need to consider two cases: $\displaystyle p\in A~\&~p\notin A$.
Both cases should force $\displaystyle p$ to be a boundary point.