Boundaries for Specific Sets with Ambient Space $\mathbb{R}$

Question

I'm trying to find the boundaries for each the following sets:

(a) $\begin{Bmatrix}\frac{1}{n}:n\in\mathbb{N}\end{Bmatrix}\overset{?}{=}\{1\}$

(b) $[0,3]\cup(3,5)\overset{?}{=}\{0,5\}$

(c) $\{r\in\mathbb{Q}:0<r<\sqrt{2}\}\overset{?}{=}\{0,\sqrt{2}\}$

(d) $\{r\in\mathbb{Q}:r\geq\sqrt{2}\}\overset{?}{=}\{\sqrt{2}\}$

(e) $[0,2]\cap[2,4]\overset{?}{=}\{2\}$

Answer 1

a) every element of the set is a boundary point
b) and (e) are correct
c) $[0,\sqrt{2}]$
d) $[\sqrt{2},\infty)$

http://mathworld.wolfram.com/BoundaryPoint.html might help. consider what the closure of these sets are as well as their complements.

Answer 2

Two of them are correct, (b) and (e); the other three are wrong. You may find it helpful to use the fact that $\operatorname{bdry}A=\operatorname{cl}A\cap\operatorname{cl}(\Bbb R\setminus A)$. In each of (a), (c), and (d) you should ask yourself first what the closure of the given set is. In (c), for instance, it’s the entire closed interval $[0,\sqrt2]$ in $\Bbb R$. Call the given set $A$. After determining what $\operatorname{cl}A$ is, you should ask yourself which points of that $\operatorname{cl}A$ are accumulation points of $\Bbb R\setminus A$; those are the points that are in the boundary of $A$. In (c), for instance, every point of $[0,\sqrt2]$ is an accumulation point of $\Bbb R\setminus A$; why?

Now see if you can use this approach to sort out (a) and (d) as well.