1. ## Closed, open, int, adh and boundary

Verify if the following sets are closed, open and find $\displaystyle \text{int}\, A,\overline A,\partial A$ and the set of accumulation points. (Does the last set have a name?)

a) $\displaystyle A=\{(x,y)\in\mathbb R^2|y>x\}$

b) If $\displaystyle a\in\mathbb R^n,$ $\displaystyle A=\{x\in\mathbb R^n|x\cdot a=1\}$

c) $\displaystyle A=\displaystyle\overline{B}\left( \left( \frac{3}{4},0 \right),\frac{1}{4} \right)\cup \bigcup\limits_{k=2}^{\infty }{B\left( \left( \frac{3}{{{2}^{k+1}}},0 \right),\frac{1}{{{2}^{k+1}}} \right)}.$

I have the definitions but I'm not sure how to proceed.

2. There is a link between continuity of maps and open and closed sets.

3. I believe that the set of limit points of $\displaystyle A$ can be called the derived set of $\displaystyle A$, and is sometimes denoted by $\displaystyle A'$.

The boundary of the first set is $\displaystyle y=x$. Note that none of the boundary points are in the set. What does this tell you about the set being open or closed?