Meaning of accumulation point | Babel Free

Definitions

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.

Examples

“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.”

“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.”

“{{quote-book|en|year=2016|author=Jonathan M. Kane|title=Writing Proofs in Analysis|pageurl=https://books.google.com.au/books?id=Hm1BDAAAQBAJ&pg=PA74&dq=%22accumulation+point%22%7C%22accumulation+points%22&hl=en&sa=X&ved=2ahUKEwjXmIa0qoPrAhVWWysKHSMKDhAQ6AEwAnoECAYQAg#v=onepage&q=%22accumulation%20point%22%7C%22accumulation%20points%22&f=false|page=74|publisher=Springer”

CEFR level