# Help with constructing closed and open set with the help of limit

I'm reading through the Basic Topology of R chapter in Abbott's analysis book just to improve my understanding after studying limits of function and sequence. I can see some connection to basic limit concept and theorem, however I find the discussion pretty thick already in terms of abstractness. I want to easily construct non-trivial example of open and closed set but it's rather difficult.

One of the exercise is this: "Give an example of an infinite collection of nested open sets whose intersection is closed and non empty." (Nested means in this case $A_1 \supset A_2 \supset A_n$)

Is this set correct $A_n=(0,\frac{n+1}{n})$? We have a sequence $\frac{n+1}{n}$ that will converges to $1$. $1$ is always in the intersection of all $A_n$. But $1$ is the limit point for the above sequence, so the intersection is closed and non-empty.

• This will only "close" one end of your interval; what about "(0"? Your example shows that a half-open interval can be generated by open intervals – user128779 Oct 30 '14 at 11:04
• Will my example work if I just set the first end point to be $\frac{1}{n}$ instead of null? – vTx Oct 30 '14 at 11:13
• Yes that would work fine since $0 \in (1/n,1 + 1/n)$ for all n > 0 – user128779 Oct 31 '14 at 23:49

$A_n=(-\frac{1}{n},\frac{1}{n})$. The intersection is $\{0\}$, which satisfies the requirement.
Your example is not correct. The intersection of your example is $(0,1]$, which is not closed.
• So the basic idea is, we want to create a sequence of set (all of them open) such that at large $n$, their intersection actually form a closed set? And the easy way to do it is to set up a sequence at their end points, which converge to one number or perhaps different number? – vTx Oct 30 '14 at 11:12
• For emphasis, any neighborhood of $0$ will contain points belonging to the intersection $\cap A_n$, so the intersection is not empty but also not closed (since $0 \not\in \cap A_n$). – hardmath Oct 30 '14 at 11:38