jwqwerty said:the definition of limit point:
a point p is a limit point of E(subset of metric space X) if every neighborhood of p contains a point q≠p which is in E.
My question is that is there a limit point p which is not in E?
SteveL27 said:Take the open unit interval E = (0,1) as a subset of the real numbers. Can you think of a real number that's not in (0,1) but that satisfies the definition of limit point of E? (Hint: Can you think of TWO such points?)
HallsofIvy said:It's actually harder to give an example of a closed set that is NOT perfect. Can you?
A limit point is a point in a set of numbers or points that can be approximated by other points in the set. In other words, it is a point where the points in the set get closer and closer to as they approach the limit point.
The formal definition of a limit point is: Let E be a subset of a metric space (X,d). A point x in X is a limit point of E if every neighborhood of x contains at least one point of E different from x itself.
A limit point is a point that can be approximated by other points in the set, whereas a cluster point is a point where there is an infinite number of points in the set that are arbitrarily close to it. In other words, a limit point is a specific point, while a cluster point is a set of points.
No, not all sets have limit points. Some sets may not have any points that can be approximated by other points in the set, while others may have an infinite number of limit points.
Yes, a limit point can be outside of the set. This means that the set does not contain the limit point, but the limit point can still be approximated by points in the set.