# Do these definitions of limit points agree?

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?

• 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’." Nov 28, 2022 at 18:26

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 $$A$$).
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’.