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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$\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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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