Limit Point of a Set: Definition, Center Point & Empty Set

[xanadu]

I've been reading Shilov's book and the definition of a limit point is as follows: x is a limit point of A if every neighborhood of x (any open ball centered at x with arbitrary radius r) contains at least one point y distinct from x which belongs to A.

I feel that from this definition a point at the center of the set would be a limit point. If that is the case then from what I understand the set B of all limit points of A is a superset of A.

However there is an exercise which says find a set A that is not empty and the set of limit points of A is empty. What could I be missing here?

 

[VeeEight]

What do you mean by 'the center of the set'?
As for your exercise, think about the integers or the naturals.

 

[slider142]

If by "a point in the center of the set", you mean, "a point that has a neighborhood of points also in the set", then that's fine. Note that if you mean a geometric center, you are assuming something that a topological set does not necessarily have, a metric, which is a function assigning the distance between two elements of the space (some topological spaces are not even metrizable!).

 

[wofsy]

I've been reading Shilov's book and the definition of a limit point is as follows: x is a limit point of A if every neighborhood of x (any open ball centered at x with arbitrary radius r) contains at least one point y distinct from x which belongs to A.

I feel that from this definition a point at the center of the set would be a limit point. If that is the case then from what I understand the set B of all limit points of A is a superset of A.

However there is an exercise which says find a set A that is not empty and the set of limit points of A is empty. What could I be missing here?

a set with the discrete topology for instance.

 

[Tac-Tics]

However there is an exercise which says find a set A that is not empty and the set of limit points of A is empty. What could I be missing here?

Reread the definition and put a huge emphasis on the word "distinct".

Maybe consider what the limit points of the following sets in R:

The empty set (even though the problem says it isn't a solution)
R itself.
The subset of the integers
The closed unit interval [0, 1]
The open unit interval (0, 1)
The half open unit intervals (0, 1] and [0, 1).
Singleton sets {0}, {1}, {e}, etc.
The subset of the rationals

 

[wofsy]

a set with the discrete topology has no limit points.

what about the converse? If a topological space has no limit points is it discrete?

 

[CaffeineJunky]

Note that the limit point does not need to be an element of the set. For example. consider the following union of intervals on the real line: (0, 1) U (1, 2). The number 1 is a limit point of this set even though it isn't an element of the set. 0 and 2 are also limit points of this set, and they lie outside of the set as well (without being in a gap). -1 is not a limit point of the set (why?).

 

[haider ali]

geometrical concept of a limit point of a set is that it is a very nearest point to that set ,means attached with that set or in other words attached with elements of that set .
haider_uop99@yahoo.com (pakistan)

 

[haider ali]

1 is limit point of A= (0,1)U(1,2) , because any open interval containing 1 contains infinite points of A .