accumulation point

"of" a subset of a topological space

• LEMMA 5.2 Let X be a Hausdorff space and A a subset of X. A point a#92;inX is an accumulation point of A if and only if a is a limit point of A.1975, Bert Mendelson, Introduction to Topology, 3rd edition, New York: Dover Publications, Inc., published 1990, →ISBN, →OCLC, §5.3, page 173:
• 2008, Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner, Elementary Real Analysis, Volume 1, Thomson-Bruckner (ClassicalRealAnalysis.com), 2nd Edition, page 153, Definition 4.9 (Closed): The set E is said to be closed provided that every accumulation point of E belongs to the set E. Thus a set E is not closed if there is some accumulation point of E that does not belong to E. In particular, a set with no accumulation points would have to be closed since there is no point that needs to be checked.