Derived Sets: Finding the Limit Points of a Set ✨

Introduction: What does a set “collect” around it?

Imagine students standing in a crowded hallway. Some students are spread out, but a few keep clustering near the same spots. In topology, a set can also have points that are “crowded” by other points from the same set. These special points are called limit points, and the collection of all of them is called the derived set. 🧠

In this lesson, you will learn how to:

explain what a derived set is,
identify limit points in examples,
connect derived sets to closure, interior, and boundary,
and use derived sets to understand how a set behaves in a topological space.

The key idea is simple: a derived set tells us where a set keeps coming back near itself, even if the point itself is not inside the set.

What is a derived set?

Let $A$ be a subset of a topological space $X$. The derived set of $A$, written as $A'$, is the set of all limit points of $A$.

A point $x \in X$ is a limit point of $A$ if every open neighborhood of $x$ contains a point of $A$ different from $x$. In symbols, for every neighborhood $U$ of $x$,

(U \setminus \{x\}) $\cap$ A $\neq$ \varnothing.

This means that no matter how carefully you zoom in around $x$, you still find points of $A$ nearby. 📌

A helpful way to think about it:

$x$ is not just “near” $A$,
it is a point where $A$ accumulates.

Example in the real line

Consider the set

A = $\left\{$$\frac{1}{n}$ : n $\in$ \mathbb{N}$\right\}$ $\subset$ \mathbb{R}.

The points $\frac{1}{n}$ get closer and closer to $0$. The number $0$ is not in $A$, but every interval around $0$ contains infinitely many points of $A$. So $0$ is a limit point of $A$.

Thus, $0 \in A'$.

Are there any other limit points? No. Each point $\frac{1}{n}$ is isolated, because you can choose a small open interval around it that contains no other points of $A$. So

$A' = \{0\}.$

This is a classic example of a derived set. ✅

How to recognize limit points

To test whether a point $x$ is a limit point of $A$, students should ask: “Does every neighborhood of $x$ contain another point of $A$?”

There are two important details:

The point $x$ itself does not have to belong to $A$.
The neighborhood must contain a point of $A$ different from $x$.

That second detail matters. If $x$ is in $A$ but isolated from the rest, then $x$ is not a limit point.

Example: a finite set

Let

A = \{2, 5, 9\} $\subset$ \mathbb{R}.

Each point can be separated from the others by a small open interval. So no point is a limit point of $A$.

Therefore,

$A' = \varnothing.$

This shows an important fact: finite sets in $\mathbb{R}$ have no limit points.

Example: an interval

Let

$A = (0,1).$

Every point inside $[0,1]$ is a limit point of $A$.

If $x \in (0,1)$, then every neighborhood of $x$ contains other points of $(0,1)$.
If $x = 0$ or $x = 1$, every neighborhood still contains points of $(0,1)$.
If $x < 0$ or $x > 1$, then there are neighborhoods of $x$ that miss $A$ entirely.

$A' = [0,1].$

Notice that the derived set can be larger than the original set. That is one reason it is closely tied to closure. 🔍

Derived set and closure

The closure of a set $A$, written $\overline{A}$, is the set of all points of $A$ together with all its limit points.

That gives the important relationship:

$\overline{A} = A \cup A'.$

This formula is one of the most useful facts in this topic.

Why this is true

Points already in $A$ are included in the closure.
Any point that can be approached by points of $A$ should also belong to the closure.

So the closure adds the missing accumulation points to the set.

Example

For

A = $\left\{$$\frac{1}{n}$ : n $\in$ \mathbb{N}$\right\}$,

we found that

$A' = \{0\}.$

Therefore,

$\overline{A} = A \cup \{0\}.$

This means the closure fills in the “gap” at $0$.

Derived sets also help explain dense sets. A set $A$ is dense in $X$ if

$\overline{A} = X.$

Using the closure formula, this means $A$ plus its limit points cover all of $X$. In other words, a dense set is spread everywhere in the space. 🌍

Derived set, interior, and boundary

The interior of $A$, written $A^\circ$, is the set of points where a whole open neighborhood stays inside $A$.

The boundary of $A$, written $\partial A$, is the set of points where every neighborhood meets both $A$ and its complement $X \setminus A$.

Derived sets connect to both of these ideas.

Relation to interior

If a point lies in the interior of $A$, it is not usually a limit point in the sense of being “forced” by surrounding points? Actually, interior points can still be limit points if there are other points of $A$ nearby. For example, every point of $(0,1)$ is a limit point of $(0,1)$.

So students should remember:

being an interior point does not mean “not a limit point,”
it only means there is a small neighborhood entirely inside the set.

Relation to boundary

Boundary points often become limit points of the set, especially if the set touches them from inside. In fact, for many common sets in $\mathbb{R}$, the boundary points are limit points.

For example, if

$A = (0,1),$

then

$\partial A = \{0,1\}.$

Those points are also limit points of $A$.

However, not every limit point is a boundary point in every setting. To see the difference, consider the whole space $X$. Then every point may be a limit point of $X$ depending on the space, but the boundary of $X$ is empty because there is no outside of $X$ inside the space.

So derived sets and boundaries are related, but they are not the same concept. ✅

More examples and patterns

Example: integers in $\mathbb{R}$

Let

$A = \mathbb{Z}.$

Each integer is isolated from the others by a neighborhood small enough to avoid other integers. So there are no limit points in $\mathbb{R}$.

Thus,

$A' = \varnothing.$

This is a good example of a set that is closed but has no derived points.

Example: rationals in $\mathbb{R}$

Let

$A = \mathbb{Q}.$

Every real number is a limit point of $\mathbb{Q}$, because rational numbers are everywhere in $\mathbb{R}$. So

$A' = \mathbb{R}.$

This shows that the derived set can be very large. Since the closure of $\mathbb{Q}$ is also $\mathbb{R}$, we have a dense set.

Example: a sequence plus its limit

Let

A = $\left\{0$$\right\}$ \cup $\left\{$$\frac{1}{n}$ : n $\in$ \mathbb{N}$\right\}$.

The only limit point is still $0$, because the points $\frac{1}{n}$ accumulate at $0$ and nowhere else. So

$A' = \{0\}.$

This example is useful because it shows that a point can belong to a set and also be a limit point of that set.

Why derived sets matter

Derived sets help describe the shape of a set in a topological space. They tell us where points cluster, where closures need extra points, and how a set interacts with its neighborhood structure.

They are especially important because they help students reason about:

closure: $\overline{A} = A \cup A'$,
dense sets: sets whose closure is the whole space,
boundary behavior: points where a set meets its outside,
isolated points: points of $A$ that are not limit points.

If a point is in $A$ but not in $A'$, it is called an isolated point of $A$. For example, in

$\left\{\frac{1}{n} : n \in \mathbb{N}\right\},$

each point is isolated.

Conclusion

The derived set $A'$ is the set of all limit points of $A$. It captures where a set keeps appearing in every neighborhood, even when the point itself is not inside the set. This makes derived sets a central idea in topology. They connect directly to closure through $\overline{A} = A \cup A'$, help explain dense sets, and relate to boundary and interior ideas. When students can identify limit points, students has a strong tool for understanding how sets behave in topological spaces. 🌟

Study Notes

The derived set $A'$ is the set of all limit points of $A$.
A point $x$ is a limit point of $A$ if every neighborhood of $x$ contains a point of $A$ different from $x$.
The closure formula is

$ \overline{A} = A \cup A'.$

Finite subsets of $\mathbb{R}$ have empty derived set.
The set $\left\{\frac{1}{n} : n \in \mathbb{N}\right\}$ has derived set $\{0\}$.
The rationals $\mathbb{Q}$ have derived set $\mathbb{R}$ in the usual topology.
A point in $A$ that is not a limit point is called an isolated point.
Derived sets help explain closure, dense sets, interior, and boundary.