accumulation point

Noun

accumulation point (plural accumulation points)

1. ("of" a subset of a topological space) Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S.
   • 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
   • 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.2008, Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner, Elementary Real Analysis, Volume 1, Thomson-Bruckner (ClassicalRealAnalysis.com), 2nd Edition, page 153

   Synonyms:

   cluster point, limit point