Confused about the definition of a limit point

Question

I am studying topology, and the definition of a limit point on my book is:

Let $A$ be a subset of a topological space $X$ and $x$ a point of $X$, we say that $x$ is a limit point of $A$ if every neighborhood of $x$ intersects $A$ at some point other than $x$ itself.

The definition of a neighborhood is an open subset containing $x$.

My question is, why do we require the intersection to be at some point other than $x$ itself?

Answer 1

The reason is that if we didn't have the phrase "other than $x$ itself", then every point of $A$ would automatically be a limit point, which is not the desired situation.

The concept of a limit point is meant (among other things) to distinguish them from isolated points of $A$. For example, if $X=\Bbb R$ and $A=(0,1) \cup \{2\}$, the point $x=2$ is qualitatively different as a member of $A$: there is no sequence of other points of $A$ that converges to $2$. The book's definition of "limit point" gives us a way of describing this phenomenon. (Check that the set of limit points of this $A$ is $[0,1]$.)

Answer 2

A point being a limit point of some set $A$ means that there is an infinite sequence of points in $A$ that approaches said point. Let $X=\mathbb{R}$ and $A= (0,1) \cup \{2\}$. $1$ is a limit point of $A$ because every neighborhood of $1$ intersects the open interval $(0,1)$ and we can use this to construct a sequence. Loosely, as you take smaller and smaller neighborhoods, you can pick a point in the intersection of the neighborhood and $A$ and this should give you a sequence of points in $A$ that do actually approach your limit point (in the topological sense).

But this idea falls flat (or at least becomes uninteresting) if you drop the condition that the intersection of the neighborhoods and $A$ contain a point other than the limit point. For example, take the point $2$, although every neighborhood of $2$ contains an element of $A$, namely $2$, the sequence we get from this is $2,2,2,\ldots$ which, in many senses, is not meaningful.

Answer 3

I would like to explain this concept in Real numbers. There are two terms, one is the limit point of a set and the other is the Adherent point.

Let $X=\mathbb{R}$ and $A \subseteq \mathbb{R}$

Limit point: A real number $x$ is limit point of $A$ if every open neighbourhood containing $x,$ conatins atleast one point $A$ other than $x$.

Example $A=[0,1] \cup \{2,3,5,7\}$ then limit points of $A$ is $[0,1]$

Adherant points: A real number $x$ is said to be adherent of $A$ if every open neighborhood of $x$ containing $x$, contains atleast one point of $A.$

$A=[0,1] \cup \{2,3,5,7\}$ then adherent points of $A$ is $[0,1] \cup \{2,3,5,7\}$

See the difference in both of these definitions, For limit point there are always infinite numbers of elements that should be present no matter how small the neighborhood you choose that is why "other than $x$ " which may not in case for adherent points

The limit point is an adherent point but not conversely.

Hopefully, I give the answer to what you are asking.

Answer 4

$\newcommand{\Ball}[3][]{B_{#2}^{#1}(#3)}\newcommand{\Del}[2]{\Ball[\times]{#1}{#2}}$There are good answers already; posting in the hope this conceptual taxonomy of the situation of a point relative to a subset is helpful to posterity.

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Fix a metric space $(X, d)$, a subset $A$ of $X$, and a point $x$ of $X$. If $r$ is a positive real number, let $\Ball{r}{x}$ denote the $d$-ball of radius $r$ about $x$, i.e., the set of $x'$ in $X$ such that $d(x, x') < r$.

Precisely one of the following occurs:

1. There exists a positive real $r$ such that $\Ball{r}{x} \subset A$. (In this case we say $x$ is an interior point of $A$.)
2. There exists a positive real $r$ such that $\Ball{r}{x} \cap A = \varnothing$, i.e., $\Ball{r}{x} \subset (X \setminus A)$. (In this case we say $x$ is an exterior point of $A$, i.e., an interior point of the complement $X \setminus A$.)
3. For every positive real $r$, both intersections $\Ball{r}{x} \cap A$ and $\Ball{r}{x} \cap (X \setminus A)$ are non-empty. (In this case we say $x$ is a boundary point of $A$. Because these three conditions are a logical trichotomy, every set $A$ has the same boundary as its complement $X \setminus A$.)

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Further, let $\Del{r}{x} = \Ball{r}{x} \setminus \{x\}$ denote the deleted $d$-ball of radius $r$ about $x$, i.e., the ball $\Ball{r}{x}$ with $x$ removed. Conceptually, just as an open ball "contains all points of $X$ sufficiently close to $x$," a deleted ball "contains all points of $X$ sufficiently close to, but distinct from, $x$."

With notation as above, precisely one of the following occurs:

1. There exists a positive real $r$ such that $\Del{r}{x} \cap A = \varnothing$. (If $x \in A$, we say $x$ is an isolated point of $A$. Otherwise $x$ is an exterior point of $A$, as above.)
2. For every positive real $r$, the intersection $\Del{r}{x} \cap A$ is non-empty (In this case we say $x$ is a limit point of $A$, whether or not $x \in A$.)

Answer 5

The most basic situation in which that definition is useful is one you are probably already familiar with: computing limits of real-valued functions.

If $E\subseteq\mathbb R$, $f:E\to\mathbb R$ is a function, $c$ is a limit point of $E,$ and $l\in\mathbb R,$ then $f(x)\to l$ as $x\to c$ means: for every $\varepsilon>0$, there is a $\delta>0$ such that, for all $x\in E,$ if $0<|x-c|<\delta$ then $|f(x)-f(c)|<\varepsilon.$

Notice that this definition applies in a very general context. For instance, it allows us to say that $\frac{\sin(1/x)}{\sin(1/x)}\to 1$ as $x\to 0,$ even though the function in question is undefined for values of $x$ which are arbitrarily close to $0$. Although we could define limits whenever $c\in\mathbb R$, this has an undesirable side-effect. If $c$ is not a limit point of $E$, then there is a $r>0$ such that $(c-r,c+r)$ does not contain any points of $E$, except possibly $c$. In that situation, for every $l\in\mathbb R$, it is vacuously true that $f(x)\to l$ as $x\to c$, and so we lose the uniqueness of limits.

In summary, the definition of a limit point is useful because it leads to us a very general definition of limits, but not so general as to be unusable.