I am having a hard time with a proof:

Prove that the set of limit points of a set is closed.

At first it seemed easy and from a logical standpoint it makes sense but I am not sure where to start.

Printable View

- Nov 12th 2007, 06:41 AMjlindboLimit point proof
I am having a hard time with a proof:

Prove that the set of limit points of a set is closed.

At first it seemed easy and from a logical standpoint it makes sense but I am not sure where to start. - Nov 12th 2007, 09:00 AMJhevon
what definition are you using for a "limit point"? a "closed set"?

strategies you may want to explore:

- show that the compliment of the set of limit points is open (this is perhaps the easiest way)

- show that if S is the set of limit points, it will contain all the limits of its convergent sequences (consisting of points in S)

- recall also that x is a limit point of a set S if it lies in the closure of S - {x}. And show that this set is closed. (this amounts to showing that the closure of a set is closed, i think)

- you can also show that for any point in S (the set of limit points of a set) that has the property that any open ball centered at that point intersects S and S compliment (that is, it is a boundary point of S) will be contained in S. (we know that such points exists based on one of the definitions for a limit point)*. thus we would show that S contains all its boundary points and is hence closed

*)**Definition:**Let*S*be a subset of a topological space*X*. We say that a point*x*in*X*is a**limit point**of*S*if every open set containing*x*also contains a point of*S*other than*x*itself. - Nov 12th 2007, 09:56 AMThePerfectHacker
I think that "limit point" here means the let of all accumulation points.

Here