- The Set A = [2,3) is closed or open
- {3} $\subset$ $\partial$A
- (X,d) is a complete metric space
Please, help me if im in the right way?
$A$ will be an open set iff, $\forall\epsilon>0$ $\exists$$B_\epsilon(x)$, with $x$ $\in$ $A$; $B_\epsilon(x)\subset$ $A$. But when $\epsilon=1$ i can find a ball that contains points in $A$ ({2}) and in $X-A$ ({1}), so is not open. Then, it also is not a closed set because exists a limit point that is not in the set ({3})
To {3} $\subset$ $\partial$A, a ball in center {3} with any radius will have point in the set and in its complement, what is false, because after the point {3} there's no more points in $X-A$.
To be Complete, all cauchy sequences might have it limit points in the set, but i can find a cauchy sequence which its limit is not. $$x_n:=\{3-(\frac{1}{n});n\in\mathbb{N}\}$$ is a cauchy sequence and its limit is {3} which is not in the set.
Am i doing fine?