HallsofIvy said:What definition of "closed set" are you using? There are several different but equivalent definitions of "closed set"- and in mathematics, proofs often use the specific words of definitions. No one can help you prove this without knowing what definition you are using.
HallsofIvy said:If you do not know the definition of "closed set", then you cannot possibly prove anything about them! Examples will not provide proofs. Since you have a specific definition of "closed interval" you might be able to prove that a closed interval contains all its accumulation points. What is your definition of "accumulation point"?
A set contains closed set accumulation points if it contains all of its limit points.
Limit points, also known as accumulation points, are points in a set where every neighborhood of the point contains at least one point from the set.
Closed set accumulation points are points that are contained within the set, while open set accumulation points are points that are not necessarily contained within the set but are still limit points.
Having closed set accumulation points ensures that the set is complete and that all of its limit points are included. This is important in many areas of mathematics, including real analysis and topology.
Yes, a set can contain both open and closed set accumulation points. For example, the set [0,1] contains both 0 and 1 as closed set accumulation points, while all other points are open set accumulation points.