Continuity via Inverse Images

students, in topology, continuity is not first defined by “drawing without lifting your pencil.” Instead, it is described using sets and how they behave under a function 🌍. This lesson explains the key idea: a function is continuous when the inverse image of every open set is open. By the end, you should be able to recognize this definition, use it in examples, and see why it matters for homeomorphisms.

Learning goals

In this lesson, you will:

• explain what inverse images are and why they matter in topology,
• use the open-set definition of continuity,
• test whether a function is continuous by checking inverse images,
• connect continuity to homeomorphisms and topological structure.

The big idea: continuity through open sets

In everyday math, a function is often called continuous if small changes in input cause small changes in output. That idea still matters in topology, but topology uses a more flexible language based on open sets.

Suppose $f : X \to Y$ is a function between topological spaces. The topology on $Y$ tells us which sets are open. The continuity condition says:

$$f \text{ is continuous} \iff f^{-1}(U) \text{ is open in } X \text{ for every open set } U \subseteq Y.$$

Here $f^{-1}(U)$ means the inverse image or preimage of $U$ under $f$. It is the set of all points in $X$ that map into $U$:

$$f^{-1}(U) = \{x \in X : f(x) \in U\}.$$

Important detail: $f^{-1}(U)$ does not require $f$ to be one-to-one, and it is not the same as an inverse function. It is simply a set of inputs that land inside $U$.

Why this works so well is that open sets describe the “local neighborhoods” of a space. If the preimage of every open set is open, then the function does not create sharp breaks in the topological sense ✨.

Understanding inverse images with examples

Let’s build intuition with a simple real-world picture. Imagine $X$ is a map of your town and $Y$ is a weather map of temperature zones. If $f$ assigns to each location in town its temperature, then for an open temperature region $U$ in $Y$, the set $f^{-1}(U)$ is the collection of places in town with temperatures in that range. Continuity means these regions stay open in the town map too.

Example 1: A familiar function on the real line

Consider $f : \mathbb{R} \to \mathbb{R}$ given by

$$f(x) = x^2.$$

Take the open set $U = (-1, 4)$ in the codomain. Then

$$f^{-1}(U) = \{x \in \mathbb{R} : x^2 \in (-1, 4)\}.$$

Since $x^2 \ge 0$ for every $x$, this becomes

$$f^{-1}(U) = \{x \in \mathbb{R} : x^2 < 4\} = (-2, 2).$$

This is open in $\mathbb{R}$. In fact, for every open set $U \subseteq \mathbb{R}$, the preimage $f^{-1}(U)$ is open. So $f$ is continuous.

Example 2: A discontinuous function

Define $g : \mathbb{R} \to \mathbb{R}$ by

$$g(x) = \begin{cases}

0, & x < 0, \\

$1, & x \ge 0.$

$\end{cases}$$$

Let $U = \left(\frac{1}{2}, \frac{3}{2}\right)$, which is open in $\mathbb{R}$. Then

$$g^{-1}(U) = [0, \infty).$$

But $[0, \infty)$ is not open in $\mathbb{R}$. So $g$ is not continuous. This example shows how the inverse-image test detects a “jump” 🧭.

Why the inverse-image definition is powerful

The inverse-image definition is one of the most useful ways to define continuity because it works in any topological space, not only in $\mathbb{R}$. That means it applies to geometric objects, discrete spaces, product spaces, quotient spaces, and many other settings.

It is also easier to use than the direct definition in many cases. Instead of checking a limit at every point, you can check how open sets behave. In topology, open sets often reveal the structure of a space more clearly than distances do.

A few important facts follow directly from this definition:

1. The inverse image of an open set under a continuous function is open.
2. The inverse image of a closed set under a continuous function is closed, because complements of open sets are closed.
3. A function is continuous if and only if the inverse image of every closed set is closed.

These statements are equivalent descriptions of the same concept.

Example 3: Continuity in a discrete space

If $X$ is a discrete topological space, then every subset of $X$ is open. For any function $f : X \to Y$, the set $f^{-1}(U)$ is always a subset of $X$, so it is automatically open. Therefore, every function from a discrete space is continuous.

This shows that continuity depends on the topology, not just on the formula for the function. The same formula may be continuous in one setting and not in another.

Equivalent characterizations and how they connect

The inverse-image definition is the standard topological definition of continuity. But in familiar spaces like $\mathbb{R}$, it connects to the usual epsilon-style idea from calculus.

For functions between metric spaces, continuity can also be described by neighborhoods or sequences. For example, if $f : X \to Y$ is continuous and $x_n \to x$ in $X$, then often $f(x_n) \to f(x)$ in $Y$ when the spaces are nice enough, such as metric spaces. However, topology uses open sets because this approach works more generally.

The open-set version is especially helpful because it is stable under common operations:

• the composition of continuous functions is continuous,
• the identity function is continuous,
• constant functions are continuous.

For example, if $f : X \to Y$ and $h : Y \to Z$ are continuous, then for every open set $W \subseteq Z$,

$$(h \circ f)^{-1}(W) = f^{-1}\big(h^{-1}(W)\big).$$

Since $h^{-1}(W)$ is open in $Y$ and then $f^{-1}\big(h^{-1}(W)\big)$ is open in $X$, the composition $h \circ f$ is continuous.

This formula is one of the most useful tools in topology because it turns a complicated map into smaller pieces.

Connection to homeomorphisms

A homeomorphism is a function that preserves topological structure perfectly. Formally, a map $f : X \to Y$ is a homeomorphism if:

• $f$ is bijective,
• $f$ is continuous,
• $f^{-1}$ is continuous.

So continuity via inverse images is central to homeomorphisms. To show two spaces are homeomorphic, you must show that open sets are preserved in both directions.

Think of two shapes made of rubber. If one can be stretched into the other without tearing or gluing, they may be homeomorphic. A circle and an ellipse are homeomorphic, but a circle and a line segment are not, because one has a “hole” behavior that the other does not. Homeomorphisms are the precise topology version of “same shape” 🔄.

The inverse-image definition helps here because it tells us what properties are preserved under homeomorphisms. Since both $f$ and $f^{-1}$ are continuous, open sets and closed sets correspond in a controlled way. That means topological properties like connectedness and compactness are often studied through continuous maps.

Example 4: A homeomorphism on an interval

Consider $f : (0,1) \to \mathbb{R}$ given by

$$f(x) = \ln\!\left(\frac{x}{1-x}\right).$$

This function is bijective, continuous, and its inverse

$$f^{-1}(y) = \frac{e^y}{1+e^y}$$

is also continuous. So $f$ is a homeomorphism. The open intervals in $(0,1)$ correspond to open sets in $\mathbb{R}$ through inverse images and images under $f$.

How to test continuity in practice

When students checks continuity using inverse images, a good method is:

1. Identify an open set $U$ in the codomain.
2. Compute $f^{-1}(U)$.
3. Decide whether $f^{-1}(U)$ is open in the domain.
4. If this works for every open set $U$, the function is continuous.

For functions on $\mathbb{R}$, you can often test intervals because open sets are built from open intervals. For abstract spaces, use the topology directly.

A useful warning: to prove a function is not continuous, you only need one open set $U$ such that $f^{-1}(U)$ is not open. One counterexample is enough.

Conclusion

Continuity via inverse images is one of the most important ideas in topology. It gives a clean and flexible definition: a function is continuous exactly when the preimage of every open set is open. This perspective works far beyond the real numbers, and it is the foundation for understanding homeomorphisms and structure-preserving maps. students, if you remember only one idea from this lesson, remember this: topology studies how spaces behave under functions by watching what happens to open sets.

Study Notes

• A function $f : X \to Y$ is continuous if and only if $f^{-1}(U)$ is open in $X$ for every open set $U \subseteq Y$.
• The inverse image is $f^{-1}(U) = \{x \in X : f(x) \in U\}$.
• The inverse image is not the same as an inverse function.
• Continuity can also be characterized using closed sets: $f^{-1}(C)$ is closed whenever $C$ is closed.
• The inverse-image definition works for all topological spaces, not just metric spaces.
• The composition of continuous functions is continuous, and this follows from inverse images.
• A homeomorphism is a bijection $f$ such that both $f$ and $f^{-1}$ are continuous.
• To show a function is not continuous, find one open set whose inverse image is not open.
• Topology focuses on structure preserved by continuous maps, especially open sets and homeomorphisms.
• The inverse-image viewpoint is a core tool for understanding continuity in the broader topic of Continuity and Homeomorphisms.