a) Consider the real line $\mathbb{R}$. The $\epsilon$-neighborhood of a point $x_0\in\mathbb{R}$ is defined as $I_{\epsilon}(x_{0}) = (x_{0} – \epsilon, x_{0} + \epsilon)$. For a set E $\subset\mathbb{R}$ define interior, exterior, and boundary points. Give the definition of open and closed sets.
b) Give a constructive description of all open subsets of the real line. Prove your statement.
So in part (a), I'm not sure how the subset E relates to $I_{\epsilon}(x_{0})$. And by the wording I am not sure if I need to define open and closed sets using the subset E.
In part (b), I am not sure what a constructive description is and much less how to prove it.
I am not sure if this helps but it is for a complex analysis class even though it's only referring to the real line.
Best Answer
Let $(X,\mathcal{T}) $ be a topological space.
General definition of open set $$ A\text{ open}\iff A\in\mathcal{T} $$ General definition of closed set $$ A\text{ closed}\iff A^c\in\mathcal{T}. $$ Now, in your case $X=\mathbb{R}$ and $\mathcal{T}$ is (supposedly) the topology induced by the standard metric on $\mathbb{R}$, $$ d(x,y)=|x-y|\quad x,y\in \mathbb{R} $$ and this defines the a set $A$ to be open if $$ \forall x\in A\ \exists \epsilon>0 \text{ s.t. } I_\epsilon(x)\subset A\iff A\in\mathcal{T}\iff A\text{ open}. $$ From this you should sort out part (b).
Now note how similar this definition with the open(!) interval $I_\epsilon(x)$ and the following definition of interior are.
Definition of interior of $A\subset X$,
$$ Int(A)=\{x\in X: \exists U\in\mathcal{T}\text{ s.t. }x\in U\text{ and } U\subset A\} $$ Definition of the closure of $A\subset X$,
$$ Cl(A)=\{x\in X: \forall U\in\mathcal{T}\text{ s.t. }x\in U\quad U\cap A\neq\emptyset\} $$ (Shortcut-)Definition of the boundary of $A\subset X$,
$$ \partial(A)=Cl(A)\backslash Int(A) $$ For part (a) I already gave you the definition of open and closed, you can now give the definitions of closure, interior and boundary of $E\subset\mathbb{R}$ translating the general ones I gave you.
(Hope it helps, consider that if you do not know the basics of topology my answer might be hardly readable)