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I'm studying if the book Multidimensional Real Analysis by Duistermaat and the definition of cluster point is:

A point $a \in \mathbb{R}^n$ is said to be a cluster point of a subset $A$ if for every $\delta >0$ we have $B(a; \delta) \cap A \neq \emptyset$, where $B(a; \delta) = \{x \in \mathbb{R}^n \;|\; ||x-a||<\delta\}$

But in many other books and internet says that:

A point $a \in \mathbb{R}^n$ is said to be a cluster point of a subset $A$ if for every $\delta >0$ we have $(B(a; \delta)-{a}) \cap A \neq \emptyset$, where $B(a; \delta) = \{x \in \mathbb{R}^n \;|\; ||x-a||<\delta\}$

It's easy to see that it isn't equivalent definitions. For example, by the first definition, the point $0$ is a cluster point of the set $S = \{0\}\cup[1,2]$, but it is not by the second one.

Which definition is the usual?

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    $\begingroup$ I think the author speaks of a cluster point to mean either a limit point or an adherent point, so that, accordingly, the definition of closure becomes simply the set of all cluster points, instead of the set itself union the set of all limit points, since every point in the set itself is an adherent point. $\endgroup$
    – Yes
    Sep 8, 2015 at 4:41
  • $\begingroup$ This irks me. "Cluster" and "accumulation" are very visual words that give the picture of several things clustering up or accumulating next to something else. "Adheres" might give the picture of a fly on a wall or something glued to something else, but I think most people would somewhat begrudgingly accept that things adhere to themselves. Thus using adherence points to include isolated points doesn't bother me as much as cluster points. $\endgroup$
    – user123641
    May 4, 2018 at 18:35

3 Answers 3

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Indeed the definitions aren't equivalent. I always saw the terms accumulation point (or limit point), and adherence point for those definitions, respectively. In simple terms, a point is adherent to a set if it is a limit point that is not isolated. My approach would be to follow the definition that each specific book uses.

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I think that the latter definition is much more usual. But if you use "adherent point" or "closure point" for the former, you are safe (I think that they are not ambiguous). Usually one calls the latter "accumulation point" or "limit point" or "cluster point", but some people might use "limit point" or (rarely) "cluster point" for an adherent point. Some use "accumulation point" for an $\omega$-accumulation point (there is no difference in $\mathbb R$ or other Hausdorff or T1 spaces). So is there any completely unambiguous term for the latter definition?

Even more complicated: Set $x_n=3$ for each $n$. Then $3$ is an accumulation point (cluster point) and even a limit point of the sequence $(x_n)_{n\in\mathbb N}$ but not an accumulation point (limit point) of the set $\{x_n\}_{n\in\mathbb N}=\{3\}$ (just an adherent point of it). So the limit/accumulation/cluster point of a sequence is a different definition that that of a set.

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A great question. It seems to me in the optimization literature, the cluster point definition adopted in Multidimensional Real Analysis by Duistermaat is very common, which is often called limit point. See the book Nonlinear Programming by Bertsekas. Is that definition completely dominant in the field? I don't know and I wish somebody in that field can help. In the math literature, the second definition is more common.

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  • $\begingroup$ Bartsekas says that $x$ is a limit point of $\{x^{k}\}$ if $x$ is a limit of some subsequence of $\{x^{k}\}$. $\endgroup$ Apr 16, 2021 at 17:17

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