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I have seen the following two versions to define what a limit point is. The first one is from Hatcher on page $5$ (note that I summarized the definition and didn't copy it, since it would have been too long): $(1)$ Let $X$ be a topological space and $A \subseteq X$. Then a point $x \in X$ is called limit point of $A$, if every open $O$ with $x \in O$ meets $A$ (that should mean that $O \cap A \neq \emptyset).$

The second one is from Wikipedia: $(2)$ Let $S$ be a subset of a topological space $X$. A point $x \in X$ is called limit point of $S$, if every neighbourhood of $x$ contains at least one point of $S$ different from $x$ itself.

Hatcher and Wikipedia both define neighbourhood the same way, that is, a neighbourhood of a point $x$ in $X$ is a set $A$ such that an open set $O$ exists with $x \in O \subseteq A$.

Suppose that $X$ is a space equipped with the discrete topology and containing at least two distinct points $x,y$. Suppose further that $A \subseteq X$ and that $x$ is a limit point of $A$ in the sense of Hatchers definition. Then $O:=\{x\}$ is open, contains $x$ and meets $A$ by assumption. Furthermore $O$ is an open neighbourhood of $x$ but does not contain a point apart from $x$.

Am I overseeing something or do these definitions just not coincide?

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    $\begingroup$ Hatcher acknowledges the discrepancy on page 6: "A small caution: Some authors use the term ‘limit point’ in a more restricted sense than we are using it here, requiring that every open set containing x contains points of A other than x itself. Other names for this more restricted concept that one sometimes finds are ‘point of accumulation’ and ‘cluster point’." $\endgroup$ Nov 28, 2022 at 18:26

2 Answers 2

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You are right, the definitions do not agree. Hatcher writes

the set of these limit points is called the closure of $A$.

Hatcher's definition is an unusual one. The Wikipedia definition is the standard one. Intuitively it means that limit points $x$ of $A$ (aka cluster points or accumulation points) can be arbitrarily closely approximated by points of $A \setminus \{x\}$. For points $x \notin A$ Hatcher's definition agrees with the standard one. A point $x \in A$ is a limit point of $A$ in the sense of Hatcher if

  • either $x$ is a limit point of $A$ in the standard interpretation
  • or $x$ is an isolated point of $A$ (which means that it has an open neigborhood not containing any other point of $A$).
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Hatcher wrote on p.6 as follows:

A small caution: Some authors use the term ‘limit point’ in a more restricted sense than we are using it here, requiring that every open set containing $x$ contains points of $A$ other than $x$ itself. Other names for this more restricted concept that one sometimes finds are ‘point of accumulation’ and ‘cluster point’.

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