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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".
Theorem 2.31. In the standard topology on R", 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 > N, the point x; is in U.
 
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