Thread: Open and Closed Sets

Ok, I am needing a little help with open and closed sets.

{1, 1/2, 1/3, 1/4, 1/5,....}
I am pretty sure that this is NOT closed since this accumulates to 0 but 0 is not included in the set. I also think that this is NOT open since 1 is a boundary point and an open set cannot contain a boundary point. Thus I think this one is neither open nor closed, am I correct?

(0,1)U(1,2)U(2,3)U(3,4)U...U(n, n+1)U...
This one has me confused. It seems like it would be open since it contains all open intervals, but then again it seems like it has no interior points since it just seems to go on forever the way the natural numbers do. Thus, I really do not know if it is open or not. However, I do feel like it is closed since it does not appear to have an accumulation point. Help?

{x: x²<2}
Is this the same as [-1,1]? If so then I think this is closed because every closed interval is a closed set. I also think that it is not open.

Please help me out, I would like to know what I've gotten right and what I've gotten wrong. Thanks!!

First problem: I think you're correct.

Second problem: Any union of open sets is open.

Third problem: This set is the same as the set $\displaystyle \{x|-\sqrt{2}<x<\sqrt{2}\}.$

You tell me: open, closed, or neither?

Ok so I can see why the second one is open. I still feel like it is closed as well though because I feel like it doesn't have any accumulation points, it just seems to go on forever and ever. Let me know if this is correct. I am basing this off of the fact that the natural numbers are closed by default because there are no accumulation points.

The third is open but not closed, correct?

Pick 1/2. That's an accumulation point of the set in your second problem, is it not? The set is not closed. The complement of the set is the natural numbers, which is closed.

You're correct about the third problem.

If you're looking at the reals using the usual metric (ie., the usual notion of distance), then the only sets that are both open and closed are the empty set, and the set of reals. Thus, once you show a set is open or closed, you don't need to check the other one.

It is neither open nor closed.

It is not open because there are points of the set which are not interior points (the finite points).
It is not closed because the accumulation point 0 does not belomg to the set.

Originally Posted by Hartlw

It is neither open nor closed.
It is not open because there are points of the set which are not interior points (the finite points). It is not closed because the accumulation point 0 does not belomg to the set.

To which question does this refer?

Originally Posted by Plato

To which question does this refer?

"{1, 1/2, 1/3, 1/4, 1/5,....}
I am pretty sure that this is NOT closed since this accumulates to 0 but 0 is not included in the set. I also think that this is NOT open since 1 is a boundary point and an open set cannot contain a boundary point. Thus I think this one is neither open nor closed, am I correct?"

For the second question: The union of any number of open sets is open.

For the third question, -2<x<2, which is open.

Originally Posted by Hartlw

"{1, 1/2, 1/3, 1/4, 1/5,....}
I am pretty sure that this is NOT closed since this accumulates to 0 but 0 is not included in the set. I also think that this is NOT open since 1 is a boundary point and an open set cannot contain a boundary point. Thus I think this one is neither open nor closed, am I correct?"

Yes that is correct.
In fact, each point in the set is a boundary point of the set.

Yo misquoted me. I simply answered your question about which question I was answering.

These are my replies:

First question:
It is neither open nor closed.
It is not open because there are points of the set which are not interior points (the finite points).
It is not closed because the accumulation point 0 does not belomg to the set.

For the second question: The union of any number of open sets is open.

For the third question, -2<x<2, which is open.

Originally Posted by Hartlw

Yo misquoted me. I simply answered your question about which question I was answering.

These are my replies:

First question:
It is neither open nor closed.
It is not open because there are points of the set which are not interior points (the finite points).
It is not closed because the accumulation point 0 does not belomg to the set.

For the second question: The union of any number of open sets is open.

For the third question, -2<x<2, which is open.

If you click on "reply" and answer addressing directly someone, it is likely that some people (most, I think) won't know

whom and what you are addressing. Instead, click on "reply with Quote" and thus the addressed message will appear in

your answer and all will know that.

Tonio

Originally Posted by Hartlw

Yo misquoted me. I simply answered your question about which question I was answering.

These are my replies:

First question:
It is neither open nor closed.
It is not open because there are points of the set which are not interior points (the finite points).
It is not closed because the accumulation point 0 does not belomg to the set.

For the second question: The union of any number of open sets is open.

For the third question, -2<x<2, which is open.

How did he misquote you? That was exactly what was in the post! If fact Plato said

Yes that is correct.

I am really confused.

Originally Posted by TheEmptySet

How did he misquote you? That was exactly what was in the post! If fact Plato said
I am really confused.

Sorry about the confusion. I was simply responding to Plato's question, in quotes. To end the confusion and set the record straight, these are my repliess to the original post.

First question:
It is neither open nor closed.
It is not open because there are points of the set which are not interior points (the finite points).
It is not closed because the accumulation point 0 does not belomg to the set.

For the second question: The union of any number of open sets is open.

For the third question, -2<x<2, which is open.