Does this definition of "limit point" really work

Question

I am reading Tapp's introduction to matrix groups for undergraduates. He gives the following definition of limit point of a set:

A point $p \in \mathbb R^m$ is called limit point of a subset $S \subseteq \mathbb R^m$ if there exists an infinite sequence of points in $S$ which converges to $p$.

He continues:

Any point $p \in S$ is a limit point is a limit point of $S$, as evidenced by the redundant infinite sequence $\{p,p,p,\dots\}$.

I understand this definition. If $S = [0,1] \cup \{2\}$ then $2$ is a limit point of $S$. Now let's compare it to the definition given on Wikipedia:

A point $x$ in $X$ is a limit point of $S$ if every neighbourhood of $x$ contains at least one point of $S$ different from $x$ itself.

In this case, $2$ is not a limit point of $S = [0,1] \cup \{2\}$.

My question is:

Wouldn't some of the theorems true for limit points in one definition suddenly be false in the other and vice versa? If so, why is it ok to define limit point the way Tapp defines it? (assuming the definition on Wikipedia is more commonly used)

My main problem with this is to understand how it's ok to have two conflicting defintions of the same thing.

Answer 1

I've seen both definitions used, but the one given by Wikipedia seems to be much more common. Under Tapp's definition, an isolated point (such as 2 in your example) is a limit point, whereas under Wikipedia's definition, an element of a set is either a limit point or an isolated point but not both. (In either case, a set may or may not contain all of its limit points.)

This lack of universal agreement about the definition should not cause a problem as long as each author is consistent in his/her own writing, and as long as it's clear to the reader which definition is in use.

Another familiar example of this sort of discrepancy is $\mathbb{N}$, the set of natural numbers. Depending on the author, this is may mean set of all positive integers (excluding zero), or the set of all nonnegative integers (including zero). As long as it's clear which definition is in use in a given context, this lack of agreement doesn't usually cause problems.

Beware, however, that that some theorems which are true using one definition (of limit points, in this case) may be false using another definition. "A set is closed if and only if it contains all of its limit points" is true under both definitions, but "a set is discrete if and only if it has no limit points" is true under the Wiki definition and false under Tapp's.

Answer 2

As stated, the definition corresponds to what I believe is normally called an adherent point of a set. In typical usage, $2$ in your example $[0,1]\cup\{2\}$ would be called an isolated point, while the set of limit points would be $[0,1]$.

Wikipedia also suggests the name "closure point" as an alternative to adherent point, which I think is a great idea (since the closure of a set consists of precisely the closure points) but I think it's very uncommon.