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In my time studying mathematics I have always found some subtle confusion with the definition of a limit point. I know it's possible for different definitions to yield the same results, and the issues with this might be syntactical rather than anything else, but this subtlety is really bothering me.

Given a topological space $(X, \mathcal{T})$ and a subset $A \subseteq X$, I've seen definitions that a limit point is any element $p \in X$ such that every neighborhood of $p$ intersects $A$ at some point other than $p$. But I've also seen the definition that $p$ is a limit point provided that if $U$ is a neighborhood of $p$ then $U \cap A \neq \emptyset$, where the latter can admit the possibility that $A \cap U = \{p\}$.

To illustrate a case where these competing definitions might get confusing, consider $\mathbb{R}$ with the particular point topology centered at $0$ (that is, any $A \in \mathcal{P}(\mathbb{R})$ containing $0$ is open). The interval $(1,2)$ is closed in this topology since its complement $(-\infty,1)\cup (2,\infty)$ is open. Notice however that for any $x \in \mathbb{R}$ the set $\{0,x\}$ is open in this topology. We expect that since $(1,2)$ is closed that it should contain all of its limit points. If we go with the second definition of limit point over the first, we find that if $x$ is a limit point of $(1,2)$ then we should have $\{0,x\} \cap (1,2) \neq \emptyset$, but this only occurs when $x \in (1,2)$. This shows however that $x$ has a neighborhood intersecting $(1,2)$ that doesn't contain any other point of $(1,2)$, so $x$ can't possibly be a limit point of this interval if we adhere to the first definition. Furthermore $(1,2)$ has no limit points according to the first definition.

I do want to acknowledge that the first definition is there to make sure that isolated points aren't mistaken for limit points. For example, in the Euclidean topology on $\mathbb{R}$ we would never look at the set $\{0\}\cup(1,2)$ and say $0$ is a limit point of this set. Is there something I'm missing here? Is there truly some ambiguity with these definitions or am I just not seeing it? Looking forward to the discussion.

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The choice of definition of a limit point is a matter of convention, one that different authors take to differently. What matters is that this difference in definitions does not alter the more important concept of the closure of a set.

If we take the definition of the closure of a set $A$ to be the union of $A$ and its set of limit points, then both definitions of limit point lead to equivalent definitions of closure. You can verify this for your particular example. According to the first definition, $(1,2)$ has no limit points, so it is equal to its closure; of course, we already knew this, since $(1,2)$ is closed in this topology. According to the second definition, the limit points of $(1,2)$ are $(1,2)$ only, and for the same reason: if $x$ is outside $(1,2)$, then $\{0,x\}$ is a neighborhood not intersecting $(1,2)$ even at $x$. So this time the closure is $(1,2)\cup (1,2) = (1,2)$, and again $(1,2)$ is equal to its closure. Thus everything checks out.

Because the closure of $A$ characterizes the convergence of sequences within $A$, this is a particularly useful definition in applications. But the idea of limit point is not so interesting without the corresponding idea of closure. You should use whatever definition of limit point you like (or is convenient for your circumstance), so long as you understand the closure.

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