For a set contained in the real numbersR, IfR -S is open, then S is closed.

I have no idea where to start on this!

Printable View

- Apr 3rd 2012, 06:16 PMKingdu8457Open/closed sets of Real Numbers
For a set contained in the real numbers

**R**, If**R -**S is open, then S is closed.

I have no idea where to start on this! - Apr 4th 2012, 06:13 AMSylvia104Re: Open/Closed sets of real numbers
Let $\displaystyle x$ be a real number. If $\displaystyle x\notin S,$ use the fact that $\displaystyle \mathbb R\setminus S$ is open to show that $\displaystyle x$ is not a limit point of $\displaystyle S.$ It will then follow that if $\displaystyle x$ is a limit point of $\displaystyle S$ then $\displaystyle x\in S,$ proving that $\displaystyle S$ contains all its limit points and so is closed.

- Apr 4th 2012, 03:40 PMModusPonensRe: Open/closed sets of Real Numbers
Another way is to note that R-(R-S)=S, that is S is the complement of an open set. It helps to think better, sometimes, to write X-A as A^c. You will see the DeMorgan laws quicker as well as other stuff.