Talk:Accumulation point

message: If a space has more than one point, then its topology is trivial if and only if every point is a limit point of every nonempty subset.
I'm afraid this simply isn't true; we already pondered this for a while in the Trivial topology article, and our conclusion was instead this:
If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \ S is still a limit point of S.
This is easy to see if you think about the closure-based definition of limit point. I'm replacing the statement and proof.
Derrick Coetzee 13:33, 5 Dec 2003 (UTC)
thread_title:  Trivial topology statement 

message: "infinitely many points x_n such that d(x,x_n)<\epsilon" is ambiguous when there may be infinitely many n for which x_n are equal. Probably this should be "infinitely many values of n such that d(x,x_n)<\epsilon.--Patrick 23:22, 2 June 2007 (UTC)
replies: text: I fixed this. Since, if for infinitely many n the values of x_n are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence, care should be taken with terminology so that the two are not confused. I edited the article accordingly. Any comments?--Patrick 12:30, 6 June 2007 (UTC)
thread_title: Ambiguity

message: "Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S whose limit is x; hence, x is called a limit point."
I think the "only if" part is wrong. It is true for "point of closure" (also called "adherent point"), but not for isolated points. E.g., in R, let S be the union of [0,1] and {5}. Then {5} is an isolated point; it is not a limit point (in the definition of "every nhbd meets a point in S different from it"), but is a point of closure. But obviously the constant sequence equal to 5 is a sequence in S, which converges to 5. 
What is true (and useful) is replacing S by S-{x}, i.e. requiring the sequence to consist of elements different from x itself:
"Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S-{x} whose limit is x; hence, x is called a limit point."
Urysohn (talk) 20:03, 17 June 2010 (UTC)
replies: text: Yes, that's a silly omission. Fixed.—Emil J. 11:10, 18 June 2010 (UTC)

text: The statement is wrong for general sequential spaces. It works however for Fréchet-Urysohn spaces, see here.—yadaddy_ag 08:21, 10 August 2016 (UTC)
thread_title:  Characterization of limit point in sequential space 

message: "Every finite interval or bounded interval that contains an infinite number of points must have at least one point of accumulation."
:I added this to the bottom of the opening paragraph and it was quickly deleted.  This is the key concept.  Please see Richard Courant.  
This article is actually very poorly written.  It does not include this key concept nor does it expand on it.  This concept is fundamental to calculus.  01001 (talk) 18:42, 10 March 2011 (UTC)
thread_title:  Point Of Accumulation  KEY CONCEPT 

message: "[...] every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself." 
Having S:={x} consisting of one element only and, for instance, the discrete topology on X, x is the limit of any sequence on S, respectively ALL elements of S are apparently in a neighbourhood of x, thus x should be "intuitively" a limit point. However, it fails the condition above. I think the definition is not well chosen and should include that case. --85.0.115.52 (talk) 16:46, 1 January 2014 (UTC)
replies: text: Ok, I see, the definition is chosen intentionally, as "No isolated point is a limit point of any set." --85.0.115.52 (talk) 17:01, 1 January 2014 (UTC)
thread_title:  Limit point of sets with cardinality 1 

message: I noticed that there is no mention of the derived set, the set of all limit points of a given set. Since these concepts are closely related, it seems logical to cross-reference.  — Preceding unsigned comment added by Fonsf (talk • contribs) 15:44, 15 June 2016 (UTC)
thread_title:  Derived Set 

message: The first sentence of the article currently readsIn mathematics, a limit point of a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.Could the phrase "with respect to the topology on X" be removed, to read as follows -- ?
In mathematics, a limit point of a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be "approximated" by points of S in the sense that every neighbourhood of x also contains a point of S other than x itself.
In other words, could it be left to the reader to assume that only one topology at a time is under discussion? I think the change would make the first sentence easier to read and understand. Dratman (talk) 20:42, 28 February 2017 (UTC)
thread_title:  Possible simplification of introductory sentence 

message: I suggest that this article (currently named "Limit point") be renamed "Limit point of a set" and that this page (i.e. "Limit point") become a disambiguation page. 
The reason being that the term "Limit point" is commonly used for notions that are not equivalent to "Limit point of a set" such as: "Limit point of a filter" (Bourbaki Topology 7.1), "Limit point of a net/sequence", "Limit point of a function with respect to a filter" (Bourbaki Topology 7.3). Mgkrupa 15:51, 21 June 2022 (UTC)
thread_title:  Renaming article Limit point of a set