Point Set Topology: Why A={1/n:n is Counting Number} is Not a Closed Set?

[kimkibun]

Why is it that the set A={1/n:n is counting number} is not a closed set?

We see that no matter how small our ε is, ε-neighborhood will always contain a point not in A (one reason is that Q* is dense in ℝ), thus, all the elements in A is boundary point, and we know that by definition, if bd(A)≤A, then A is closed (this is what Steven R. Lay used in his book). (≤-subset). A good friend of mine told me that A does not contain cluster point and that made A not a closed set, he said (and i know) that closed set always contain cluster points. is this some sort of contradiction?

 

[micromass]

Sure, all points in A are boundary points. This means that $A\subseteq bd(A)$. What you want is the reverse inclusion! So you have to show that all boundary points are exactly in A. This is not true here, there is a boundary point of A that is not in A.

 

[Bacle2]

Why is it that the set A={1/n:n is counting number} is not a closed set?

We see that no matter how small our ε is, ε-neighborhood will always contain a point not in A (one reason is that Q* is dense in ℝ), thus, all the elements in A is boundary point, and we know that by definition, if bd(A)≤A, then A is closed (this is what Steven R. Lay used in his book). (≤-subset). A good friend of mine told me that A does not contain cluster point and that made A not a closed set, he said (and i know) that closed set always contain cluster points. is this some sort of contradiction?

Re the cluster points, it is true that a closed set contains all its cluster points. Maybe your friend was referring to your set A: does it contain all its cluster points?

 

[HallsofIvy]

As micromass said, the fact that all points in A are boundary points is irrelevant. In order to be closed, all boundary points must be in A. Since this is a sequence of points converging to 0, 0 is as boundary point but is not in A. That is what your friend was saying.

 

[tait]

The set 1/n (n = 1,2,...) doesn't contain any limit points (can you see why?), but it certainly has a limit point (can you see what the limit point is?) and so from the definition we see that this set is not closed.