accumulation point

accumulation point (plural accumulation points)

1. (topology, "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.

   Synonyms: cluster point, limit point

   • 1975, Bert Mendelson, Introduction to Topology, 3rd edition, New York: Dover Publications, Inc., published 1990, →ISBN, →OCLC, §5.3, page 173:

     LEMMA 5.2    Let X be a Hausdorff space and A a subset of X. A point ${\displaystyle a\in X}$ 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.

   • 2016, Jonathan M. Kane, Writing Proofs in Analysis, Springer, page 74:

     A set ${\displaystyle A}$ has an accumulation point ${\displaystyle p}$ if for every ${\displaystyle \textstyle \epsilon >0}$ there is an ${\displaystyle \textstyle x\in A}$ with ${\displaystyle \textstyle x\neq p}$ and ${\displaystyle \textstyle \vert x-p\vert <\epsilon }$. Informally, ${\displaystyle p}$ is an accumulation point of ${\displaystyle A}$ if there are points of ${\displaystyle A}$ that are arbitrarily close to ${\displaystyle p}$. Note that the fact that ${\displaystyle p}$ is an accumulation point of the set ${\displaystyle A}$ has nothing to do with whether ${\displaystyle p}$ is actually an element of ${\displaystyle A}$. For example, the set ${\displaystyle \textstyle A=\{{\frac {1}{n}}\vert n\in \mathbb {N} \}}$ has one accumulation point, ${\displaystyle 0}$, because for every ${\displaystyle \textstyle \epsilon >0}$ there is an ${\displaystyle \textstyle n\in \mathbb {N} }$ with ${\displaystyle \textstyle {\frac {1}{n}}<\epsilon }$. Here the accumulation point ${\displaystyle 0}$ is not an element of the set ${\displaystyle A}$.

2. (mathematical analysis, "of" a sequence) Given a sequence s_i, a point x whose every neighborhood contains at least one element of the sequence distinct from x.

   Synonyms: cluster point, limit point

3. (systems theory, dynamical systems, chaos theory) For certain maps, a point beyond which periodic orbits give way to chaotic ones.
   • 1995, Milos Marek, Igor Schreiber, Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press, page 77:

     The chaotic set (not necessarily attracting) is formed after the first accumulation point (${\displaystyle a_{\infty }\approx 3.570}$ for the logistic mapping) is reached. In the chaotic region of the logistic map the periodicity re-emerges in periodic windows which are bounded by the accumulation point from the right and by the saddle-node bifurcation from the left. A reverse bifurcation sequence occurs above the accumulation point.

• If X is a T₁ space (a broad class that includes Hausdorff spaces and metric spaces), then the set of points in S in each neighborhood of an accumulation point x is at least countably infinite.
  • If each neighborhood's intersection with S is uncountably infinite, the term condensation point can be used. Terms such as ${\displaystyle \aleph _{0}}$-accumulation point (or ${\displaystyle \omega }$-accumulation point) and ${\displaystyle \aleph _{1}}$-accumulation point may also be used.
  • The term complete accumulation point may be used if the cardinality of the set of points in any given neighborhood of x that are also in S is equal to the cardinality of S.
• The sequence case can be regarded as a particular instance of the topological definition. For a sequence of real numbers, for instance, the topological space is the real number line (equipped with an order topology provided by the absolute value metric), of which the sequence is subset. If the sequence has a limit, it must be an accumulation point. (But note that a sequence may have more than one accumulation point.)
  • Consequently, both cases can be explained and discussed in similar mathematical language.

• isolated point

• limit point

• condensation point

• complete accumulation point

• limit set

• Accumulation point on Wikipedia.Wikipedia
• Accumulation point on Encyclopedia of Mathematics
• Accumulation Point on Wolfram MathWorld