Open and Closed sets

**Unoticed** · November 27th 2007, 07:34 AM

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.

**Plato** · November 27th 2007, 08:12 AM

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?

**ThePerfectHacker** · November 27th 2007, 08:15 AM

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, $1/2 \in B$ but if we choose $\epsilon$ small enough then $(1/2-\epsilon,1/2+\epsilon)$ does not lie in $B$ , take for example $\epsilon = .1$ .

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

**Unoticed** · November 27th 2007, 08:39 AM

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.

**ThePerfectHacker** · November 27th 2007, 08:47 AM

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 $B$ has no interior points. Think about it.

**Unoticed** · November 27th 2007, 08:51 AM

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*?

**ThePerfectHacker** · November 27th 2007, 08:58 AM

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 $n/(n+1)$ ) which is in $B^*$ is an interior point.

**Unoticed** · November 27th 2007, 09:00 AM

ah right, thanks.

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