Topology — Open Sets, Continuity & Key Concepts

Topology is the branch of math that studies spaces using open sets and continuity instead of exact distance. If you are learning topology basics, the fastest way in is this: choose which subsets are open, then define continuity from that choice.

That is why topology can compare spaces by local behavior rather than by lengths, angles, or coordinates.

What A Topological Space Is

Let $X$ be a set. A topology on $X$ is a collection $\mathcal{T}$ of subsets of $X$, called open sets, such that:

1. $\varnothing$ and $X$ are in $\mathcal{T}$.
2. Any union of sets in $\mathcal{T}$ is also in $\mathcal{T}$.
3. Any finite intersection of sets in $\mathcal{T}$ is also in $\mathcal{T}$.

The pair $(X,\mathcal{T})$ is called a topological space.

This definition is abstract on purpose. Once you know which sets are open, you can define continuity, neighborhoods, and closed sets without ever writing down a distance formula.

Why Open Sets Matter

Open sets tell you what it means for a point to have room around it. In the usual topology on $\mathbb{R}$, an open interval like $(1,3)$ is open because every point inside it has a smaller interval around it that still stays inside $(1,3)$.

That local idea is the heart of topology. A set is open when each of its points sits inside the set with some local wiggle room.

This is also why openness depends on the topology. The same subset can be open in one topology and not open in another.

How Open Sets Define Other Basic Ideas

Neighborhoods

A neighborhood of a point is any set that contains an open set around that point. It is topology's way of talking about local closeness without using distance.

Closed sets

A set is closed if its complement is open. In familiar spaces like the real line, closed sets often contain their boundary, but that is not the definition.

Continuity

If $X$ and $Y$ are topological spaces, a function $f:X \to Y$ is continuous if for every open set $U \subseteq Y$, the inverse image $f^{-1}(U)$ is open in $X$.

$$f \text{ is continuous if } f^{-1}(U) \text{ is open for every open } U \subseteq Y$$

This agrees with the usual calculus idea on $\mathbb{R}$, but it also works in spaces where distance is not the main structure.

Homeomorphism

A homeomorphism is a bijective function that is continuous and has a continuous inverse. If two spaces are homeomorphic, topology treats them as having the same shape in the topological sense.

That statement depends on continuity in both directions. A bijection alone is not enough.

Worked Example: Why $f(x)=x^2$ Is Continuous In The Usual Topology

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

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

using the usual topology on $\mathbb{R}$.

Take the open set $(1,4)$ in the codomain. Its inverse image is

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

which simplifies to

$$(-2,-1)\cup(1,2).$$

That set is open in the usual topology on $\mathbb{R}$ because it is a union of open intervals. This example shows the right direction to check: continuity is about inverse images of open sets, not about whether $f$ sends open sets to open sets.

On $\mathbb{R}$, this condition turns out to match the usual limit-based definition of continuity.

Common Mistakes With Topology Basics

Thinking "open" means the same thing everywhere

It does not. Openness is defined relative to a chosen topology.

Confusing image and inverse image

Continuity is about inverse images of open sets being open. Images of open sets do not have to be open unless you add extra conditions on the function.

Treating topology as metric geometry with new words

Many topological spaces do come from metrics, but topology itself is more general. It keeps only the structure needed to talk about local behavior and continuity.

Assuming every bijection is a topological equivalence

To be a homeomorphism, a map must be bijective, continuous, and have a continuous inverse.

Where Topology Is Used

Topology provides the language behind continuity, compactness, connectedness, and convergence in modern analysis. It also appears in geometry, dynamical systems, differential equations, and data analysis that studies shape at a coarse level.

For a beginner, the practical use is simpler: topology explains why continuity can be defined in a way that does not depend on coordinates or a distance formula.

Try A Similar Problem

Use the same function $f(x)=x^2$, but now start with the open set $(0,1)$. Compute $f^{-1}((0,1))$ and check whether it is open in $\mathbb{R}$.

If you want the next step after that, explore another case in continuity and compare the topological definition with the limit-based one from calculus.