# Thread: Open and Closed sets

1. ## Open and Closed sets

The set B is given by

B = { n/n+1 | n = a natural number } as a subset of the Real numbers. Is this set

i) Open
ii) Closed

Provide justification.

2. a) What does it mean to be an open set in the reals?
b) What does it mean to be an closed set in the reals?

3. Originally Posted by Unoticed
The set B is given by

B = { n/n+1 | n = a natural number } as a subset of the Real numbers. Is this set

i) Open
ii) Closed

Provide justification.
The set is not open. Note, $\displaystyle 1/2 \in B$ but if we choose $\displaystyle \epsilon$ small enough then $\displaystyle (1/2-\epsilon,1/2+\epsilon)$ does not lie in $\displaystyle B$, take for example $\displaystyle \epsilon = .1$.

The set is not closed. We need to show that its complement is not open. Let $\displaystyle B^*$ be its complement. Since $\displaystyle 1\not \in B$ it means $\displaystyle 1\in B^*$ but not interval $\displaystyle (1-\epsilon,1+\epsilon)$ lies completely outside $\displaystyle B$ because $\displaystyle n/(n+1)$ converges to $\displaystyle 1$.

4. Hey, thanks for that. Could you help me out with interior points. The next part of the question asks to state (if any) which are the non-interior points.

5. Originally Posted by Unoticed
Hey, thanks for that. Could you help me out with interior points. The next part of the question asks to state (if any) which are the non-interior points.
The set $\displaystyle B$ has no interior points. Think about it.

6. So all points in B are non-interior points of B, I see now why B is not open.

If B* is the complement of B, does this mean that all points in B* are non-interior points of B*?

7. Originally Posted by Unoticed
If B* is the complement of B, does this mean that all points in B* are non-interior points of B*?
Every point except for 1 (which is the limit of $\displaystyle n/(n+1)$) which is in $\displaystyle B^*$ is an interior point.

8. ah right, thanks.