Closed, open, int, adh and boundary

**Connected** · March 30th 2011, 05:41 PM

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

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

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

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

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

**girdav** · March 31st 2011, 02:51 AM

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

**DrSteve** · March 31st 2011, 04:37 AM

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

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

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