LEMMA 5.2 Let X be a Hausdorff space and A a subset of X. A point a∈X 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

Accumulation point

Meaning

Noun

Synonyms