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?
Best Answer
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.