# Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set {Xilien C A and x; – p, then p is in the closure of A. Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If
the set {Xilien C A and x; – p, then p is in the closure of A.
As we shall see later, in some topological spaces, the converse of the previous result
is not true. But it is true for R&quot;.
Theorem 2.31. In the standard topology on R&quot;, if p is a limit point of a set A, then there
is a sequence of points in A that converges to p.
Definition. Let (X, J) be a topological space, A a subset of X, and p a point in X. Then p
is a limit point of A if and only if for each open set U containing p, (U -{p)) nA # 0.
Notice that p may or may not belong to A.
Definition. A sequence in a topological space X is a function from N to X. The image of
i under this function is a point of X denoted x; and we traditionally write the sequence
by listing its images: X1, X2, X3, … or in shorter form: (x;)iEN-
Definition. A point p E X is a limit of the sequence (X;)ien, or, equivalently, (X )iEN
converges to p (written x; – p), if and only if for every open set U containing p, there
is an N E N such that for all i &gt; N, the point x; is in U.

Looking for a Similar Assignment? Order now and Get 10% Discount! Use Coupon Code “Newclient”

The post Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set {Xilien C A and x; – p, then p is in the closure of A. appeared first on Superb Professors.