Definition of cluster point 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?

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.
