Lecture
In this section we shall once again turn to the study of the properties of topological spaces and consider the operations of closure, of picking out the interior and the boundary of a set, and the concept, closely related to these operations, of limit and boundary points. All these concepts generalize familiar concepts of mathematical analysis.
Let (X, t) be a topological space.
Definition 10. The closure `A of a set A ⊂ X is called the intersection of all closed sets containing A.
The following statements are obvious.
1. The closure `A is the smallest closed set containing A.
2. If A is closed, then `A = A.
A closed set can be characterized by means of the concept of a limit point, defined below.
Definition 11. A point x ∈ X is called limit for a given set A ⊂ X, if in every neighbourhood U(x) of the point x there is at least one point x' ∈ A different from x.
Example 17. Consider in R1 the sets A = {n}, B = {1/n}, n = 1, 2,…, C = (0, 1), D = [0,1].
The set A has no limit points, the set B has one limit point, 0; the limit points of the sets C and D fill the entire interval [0, 1].
The concept of a limit point in a topological space is, as is easily seen, a generalization of the concept of a limit point in analysis. Let us prove several useful statements connected with the concept of limit points.
Theorem 4. A set A ⊂ X is closed if and only if it contains all its limit points.
Proof. Let A be closed, x – a limit point of A, and x ∉ A. Then x belongs to the open set O(x) = X\A, which is a neighbourhood of the point x. But O(x) ∩ A = ∅, which contradicts the fact that x is a limit point.
Let A contain all its limit points. Let us show that it is closed, i.e., that its complement U = X\A is open. For this, by virtue of Theorem 2, it suffices to show that for any point x ∈ U there is found a neighbourhood O(x) of the point x such that O(x) ⊂ U. Suppose the contrary: that for some point x0 ∈ U and every neighbourhood O(x0) of it there is a point x' ∈ O(x0) such that x' ∉ U. Then x' ∈ X\U = A, hence x0 is a limit point for A, and therefore x0 ∈ A, in contradiction with the assumption that x0 ∈ U = X\A.
The set of all limit points of a set A is called the derived set of the set A and is denoted A'. Thus a new operation arises, associating to every set A ⊂ X its derived set A'.
Theorem 5. For any set A ⊂ X the set A ∪ A' is closed.
Proof. Let us show that the set X\(A ∪ A') is open. Let x – be an arbitrary point of X\(A ∪ A'). Then x is not a limit point for A, so there is found an open neighbourhood O(x) of it such that O(x) ∩ A = ∅. Let x' ∈ O(x) be an arbitrary point. Then O(x) is a neighbourhood of the point x', and O(x) ∩ A = ∅. Hence x' is not a limit point for A, and O(x) ∩ A' = ∅. Thus O(x) ⊂ X\(A ∪ A'); in view of the arbitrariness of x the set X\(A ∪ A') is open, and therefore A ∪ A' is closed.
Let us prove the basic statement about the structure of the closure of a set.
Theorem 6. `A = A ∪ A' for every set A, A ⊂ X.
Proof. By Theorem 5 the set A∪A' is closed. Hence, by the definition of closure, `A ⊂ A∪A'. On the other hand, any closed set containing A contains all its limit points (see Theorem 4), and hence all the limit points of A, and therefore contains A'. It follows that A ∪ A' ⊂`A. Thus, A = A ∪ A'.
Definition 12. A point x ∈ A is called an isolated point of the set A if there exists a neighbourhood O(x) of the point x, containing no points of the set A other than x.
Obviously, a point x∈ A is isolated if and only if x ∈ A\ A'.
Definition 13. A set A is called discrete, if every one of its points is isolated.
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