# Thread: Closed set

1. ## Closed set

Can a subset in Rk be not open and not closed???

My book gives an example of 1/n as n=1,2,3..... Why is it not open and not closed? Could someone explain?

2. Originally Posted by felixmcgrady
Can a subset in Rk be not open and not closed???

My book gives an example of 1/n as n=1,2,3..... Why is it not open and not closed? Could someone explain?

If by "Rk" you mean $\displaystyle \mathbb{R}$ , then $\displaystyle \left\{\frac{1}{n}\right\}_n^\infty$ is not open because it is not true that for every point in it there exists an open interval containing the point which is completely contained in the set, and it is not closed because it doesn't contain all its accumulation points (the point zero=0), or also: it is not closed because its complement is not open (for the point x = 0 in the complement there doesn't exist an interval containing it and which is contained in the complement).

Tonio

3. In fact, any "half open" interval like [0, 1) or (-3, -2], in R, is neither open nor closed.

Something like $\displaystyle \{(x,y)| 0\le x\le 3, 0< y< 5$ in $\displaystyle R^2$ is neither open nor closed.