# Philosophical reason behind definition of limit point.

Let $(X,\mathcal{T})$ be a topological space and let $A\subseteq X$.

A point $x\in X$ is a limit point of $A$ iff for any open neighborhood $U$ of $x$, $$A\cap (U\setminus\{x\})\neq\emptyset.$$

In otherwords, any neighborhood $U$ of $x$ contains a point of $A$ (other than $x$).

The philosophical questions behind the definition, is why does the definition exclude the point $x$ (as in $U\setminus\{x\}$)?

I understand that one reason is to ensure that isolated points are not limit points, which brings us to another question, why not let isolated points be considered limit points?

Thanks, and hope this is not out of topic.

• If we didn't exclude $x$, then every point in $X$ would be a limit point of $X$. Commented Apr 17, 2016 at 7:27
• Some authors (e.g. Pugh, Real Mathematical Analysis) do include isolated points as limit points. Quote from Pugh: "Let $M$ be a metric space and $S$ be a subset of $M$. A point $p \in M$ is a limit [point] of $S$ if there exists a sequence $(p_n)$ in $S$ that converges to it.
– user169852
Commented Apr 17, 2016 at 7:37

Your definition in words: every open neighbourhood of $x$ intersects $A$ in a point different from $x$.
Then every point of $A$ itself is either a limit point of $A$ or an isolated point of $A$ (which means: for some open neighbourhood $U$ of $x$ we have $U \cap A = \{x\}$) and these are mutually exclusive. Of course outside of $A$ there can also be limit points of $A$, and there we automatically intersect $A$ in a point different from $x$, so there the $\setminus \{x\}$ part is irrelevant.
So this limit point definition nicely partitions the points of $A$ in different types, and they behave quite differently: any function on $A$ will be automatically continuous in any isolated point, and in a metric (or first countable) space any limit point $p$ will be a limit of points from $A$ that are all different from $p$, which explains the terminology: in a metric space the limit points of $A$ are the non-trivial limits (excluding all trivial sequences that have a constantly $p$ subsequence) of $A$.
A point $x$ such that $U \cap A \neq \emptyset$ is called an adherence point of $A$ (in some texts, but some others call this a limit point, so you should always check what is meant in a specific context!) and then we can say that the closure of $A$ is exactly the set of adherence points of $A$. All points of $A$ (isolated or not) are trivially adherence points of $A$ as well. For limit points we can say that the closure of $A$ is the union of $A$ and its limit points (we need all of $A$, because maybe some or all points of $A$ are isolated, but we need them to be in the closure of $A$ by definition).