Limit Points in Topology

Imagine you are standing on a beach and dropping pebbles into the sand 🏖️. No matter where you place a marker, you can often find more pebbles very close to it. In topology, this idea of being “surrounded” by points from a set leads to an important concept called a limit point. students, understanding limit points helps you connect closure, interior, and boundary in a clear and useful way.

Learning objectives:

Explain what a limit point is and why the idea matters.
Use the definition of a limit point to test examples.
Connect limit points to closure, interior, and boundary.
Describe how limit points help classify sets in topology.
Support reasoning with examples from familiar number sets.

What Is a Limit Point?

A point $x$ is a limit point of a set $A$ if every open neighborhood around $x$ contains at least one point of $A$ different from $x$ itself. In simpler words, no matter how tiny a “bubble” you draw around $x$, you always find points from $A$ inside that bubble. 🌟

The key idea is that the points of $A$ keep appearing arbitrarily close to $x$.

In a topological space, the formal definition is:

A point $x$ is a limit point of $A$ if for every open set $U$ with $x \in U$, we have $U \cap (A \setminus \{x\}) \neq \varnothing$.

This means:

$x$ does not have to belong to $A$.
Even if $x \in A$, the point $x$ must still be approached by other points of $A$.
A limit point is about closeness, not just membership.

A simple real-world picture

Think of a crowd at a concert 🎤. If a person is a limit point of the crowd, then every small area around that person contains other people from the crowd. The crowd is packed closely around them. If a point is far away from everyone else, it is not a limit point.

Testing the Definition with Examples

Let’s work in the real numbers $\mathbb{R}$ with the usual topology.

Example 1: The interval $A = (0,1)$

Consider the set $A = (0,1)$.

Every point $x$ with $0 < x < 1$ is a limit point of $A$ because any open interval around $x$ contains infinitely many points of $(0,1)$.
The endpoints $0$ and $1$ are also limit points, even though they are not in the set. For example, any interval around $0$ contains numbers like $0.1$, $0.01$, or $0.001$ from $A$.

So the set of limit points of $(0,1)$ is $[0,1]$.

This shows something important: limit points can lie inside the set or outside the set.

Example 2: The set $B = \{1, 2, 3\}$

Now consider a finite set like $B = \{1, 2, 3\}$.

None of these points is a limit point in the usual topology on $\mathbb{R}$. Why? Because you can choose a small enough open interval around $1$ that contains no other points of $B$ besides $1$ itself. The same is true for $2$ and $3$.

So finite sets in $\mathbb{R}$ have no limit points.

Example 3: The set $C = \left\{\frac{1}{n} : n \in \mathbb{N}\right\}$

This set is famous in topology. It contains the points

$$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$$

These points get closer and closer to $0$.

The point $0$ is a limit point of $C$ because every open interval around $0$ contains some $\frac{1}{n}$.

Are the points $\frac{1}{n}$ themselves limit points? No. Each one is isolated from the others by a small enough neighborhood.

So the set of limit points of $C$ is $\{0\}$.

This example is very useful because it shows how a set can have a limit point that is not in the set itself. It also shows that a set can have many points but still only one limit point.

Limit Points and Closure

Limit points are tightly connected to closure.

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

So one way to remember closure is:

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

where $A'$ is the set of limit points of $A$.

This formula is very important because it shows how limit points build the closure of a set.

Example with $(0,1)$

For $A = (0,1)$, the limit points are all points in $[0,1]$. Since the set already contains points in $(0,1)$, the closure is also $[0,1]$.

Example with $\left\{\frac{1}{n}\right\}$

For $C = \left\{\frac{1}{n} : n \in \mathbb{N}\right\}$, the only limit point is $0$. The closure is

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

This means closure “fills in” all the points that are forced by nearby points of the set.

Limit Points and Interior

The interior of a set $A$, written $A^\circ$, is the set of all points of $A$ that have some open neighborhood completely inside $A$.

Limit points and interior are different ideas:

Interior points are safely inside the set.
Limit points are points where the set keeps appearing nearby.

A point can be both an interior point and a limit point. In fact, any interior point of a nonempty open interval like $(0,1)$ is a limit point too.

But not every limit point is an interior point. For example, $0$ is a limit point of $(0,1)$, but it is not an interior point because no open interval around $0$ lies entirely inside $(0,1)$.

This helps students see why limit points are broader than interior points. Interior asks, “Is there room entirely inside the set?” Limit points ask, “Does the set keep showing up near here?”

Limit Points and Boundary

The boundary of a set $A$ consists of points where every neighborhood meets both $A$ and its complement. Boundary points are like edge points. 🧩

Limit points and boundary points are related, but they are not the same.

For a set like $(0,1)$:

The points $0$ and $1$ are boundary points.
They are also limit points.
Points inside $(0,1)$ are limit points too, but they are not boundary points.

So boundary points are special limit points that sit on the edge of the set.

A useful relationship is:

Interior points are away from the boundary.
Boundary points may or may not be limit points depending on the set.
Limit points describe accumulation, while boundary describes separation between a set and its outside.

For many common sets in $\mathbb{R}$, boundary points are limit points, but the concepts are not identical.

How to Check Whether a Point Is a Limit Point

When solving problems, use a careful step-by-step method:

Pick a point $x$.
Consider an arbitrary open neighborhood around $x$.
Ask whether that neighborhood always contains a point of the set different from $x$.
If the answer is yes for every neighborhood, then $x$ is a limit point.
If you can find one neighborhood that avoids all other points of the set, then $x$ is not a limit point.

Example: Is $2$ a limit point of $\{1,2,3\}$?

No. A small open interval around $2$, such as $(1.9, 2.1)$, contains only $2$ from the set. Since it does not contain another point of the set, $2$ is not a limit point.

Example: Is $0$ a limit point of $\left\{\frac{1}{n}\right\}$?

Yes. Any interval around $0$ contains some $\frac{1}{n}$ because the terms of the sequence get arbitrarily small. Therefore $0$ is a limit point.

Why Limit Points Matter

Limit points help explain how sets behave under closeness and approximation. They are important in several areas of topology and analysis.

They describe where a set “accumulates.”
They are part of the definition of closure.
They help distinguish isolated points from points surrounded by the set.
They support reasoning about dense sets, since dense sets have closure equal to the whole space.

For example, the rational numbers $\mathbb{Q}$ are dense in $\mathbb{R}$ because every real interval contains rational numbers. That means every real number is a limit point of $\mathbb{Q}$ or lies in its closure. This is a powerful idea in topology because it shows how a set can be thin yet still spread everywhere.

Conclusion

Limit points are one of the core ideas in topology. A point is a limit point of a set when every neighborhood around it contains another point of the set. This idea links directly to closure, since the closure of a set is the set plus all of its limit points. It also helps students understand the difference between interior points, boundary points, and accumulation behavior.

By practicing examples like intervals, finite sets, and sequences such as $\left\{\frac{1}{n}\right\}$, you can build strong intuition for how limit points work. This makes it easier to reason about closure, interior, boundary, and dense sets in a connected way.

Study Notes

A limit point of a set $A$ is a point where every open neighborhood contains a point of $A$ different from the point itself.
Limit points can be inside the set or outside it.
The closure of a set is $\overline{A} = A \cup A'$.
Interior points are points with a neighborhood completely inside the set.
Boundary points are points where every neighborhood meets both the set and its complement.
Finite subsets of $\mathbb{R}$ usually have no limit points.
The set $\left\{\frac{1}{n} : n \in \mathbb{N}\right\}$ has limit point $0$.
The interval $(0,1)$ has limit points $[0,1]$.
Dense sets have closures equal to the whole space, so limit points are central to understanding density.
Limit points describe accumulation, which is a key part of closure, interior, and boundary in topology.