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I was being taught the Limit points and Cluster points by my teacher. He mentioned that there's a difference in between them as stated in S. Kumaresan's Topology of Metric Spaces:

Taking the metric space as the set of Real numbers,

Limit points: Let $A\subseteq\mathbb{R}$. A point x $\in\mathbb{R}$ is a limit point of $A$ iff $\forall\epsilon>0, \left(x-\epsilon,x+\epsilon\right)\cap A\neq\phi$

whereas

Cluster points: A point $c$ is said to be the cluster point of a set $A\left(\subseteq\mathbb{R}\right)$ if $\forall\epsilon>0$, $\left(c-\epsilon,c+\epsilon\right)\cap A-\{c\}\neq\phi$.

A simple example would explain the difference between these two:

Let $A=\{1,2,3,4,5\}$. Then according to these definitions of limit and cluster points, the set of all the limit points of $A$ is $\{1,2,3,4,5\}$. But, there is no cluster point of this set.

But in Rudin, the limit point is defined as:

Limit points: A point $x$ is said to be a limit point of a set $A\left(\subseteq\mathbb{R}\right)$ if $\forall\epsilon>0$, $\left(x-\epsilon,x+\epsilon\right)\cap A-\{x\}\neq\phi$.

which is the same as the definition of cluster points in S. Kumaresan's book.

So, going by this definition, we find that there is no limit or cluster points of the set $A$ in my example.

I'm a little curious to know if there is any difference in the set of cluster points and limit points? And if so, then why is it not mentioned in Rudin?

(Note: The above definitions are not the exact words of the books though. The original definitions took any metric space instead of only the set of Real numbers)

There's a possible similar question posted here but I couldn't get it completely.

DeBARtha
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  • There are some variations in these definitions. Check the definitions carefully from the book you are reading. – Kavi Rama Murthy Dec 03 '20 at 05:06
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    Things are what we say the are and different text books will sometimes have different definitions ans means for words. What Kumaresan calls "limit points" Rudin has no name for and doesn't use at all. Notice all points of a set, even isolated points, are what Kumaresan calls limit points. I don't think Rudin considers this concept is useful. What Kumerasen calls "cluster points" are exactly and precisely what Rudin calls "limit points". – fleablood Dec 03 '20 at 05:57
  • Does this answer your question? Difference between cluster points and limit points? – AKP2002 Dec 25 '21 at 13:59

1 Answers1

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Being honest I always get confused by the names given to these kind of points (especially since the terms are not universal), but what there is no doubt is that for $A \subseteq \mathbb R$ we have $$\overline{A} = \{x \in \mathbb R : (x-\varepsilon,x+\varepsilon) \cap A \neq \varnothing \textrm{ for all } \varepsilon>0\}$$ and that $$A' = \{x \in \mathbb R : (x-\varepsilon,x+\varepsilon) \cap (A\setminus \{x\}) \neq \varnothing \textrm{ for all } \varepsilon>0\}.$$ Note that $x \in A'$ if and only if $x \in \overline{A \setminus \{x\}}$, and that $\overline{A} = A \cup A'$. So, if $A$ is a finite set of points, $A' = \varnothing$ and then $\overline A = A$.

azif00
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