In geometry, we often talk about distance. The distance between two points on a map, the length of a line, or the radius of a circle are all familiar concepts. But what if we wanted to describe nearness or closeness without using numbers? This is the core idea behind topology.

Topology provides a way to talk about the structure of a space by defining which points are 'near' each other. It does this not by measuring distance, but by specifying a collection of 'open sets'.

Imagine a set of points, which we'll call $X$. A topology on $X$ is a collection of subsets of $X$, which we'll call $\mathcal{T}$. For $\mathcal{T}$ to be a topology, its members (the open sets) must follow three simple rules.

A set $X$ combined with a topology $\mathcal{T}$ is called a topological space, written as $(X, \mathcal{T})$. This simple framework allows us to study the essential properties of shapes that don't change when they are stretched or bent.

The sets that belong to the topology $\mathcal{T}$ are called open sets. This is the fundamental building block. From this, we can define what it means for a set to be closed.

A subset of $X$ is called closed if its complement is an open set. The complement is just everything in $X$ that isn't in the subset. So, if a set $U$ is open, then everything else, $X \setminus U$, is closed. Likewise, if a set $C$ is closed, its complement $X \setminus C$ is open.

It's important to remember that 'closed' is not the opposite of 'open' in topology. A set can be open, closed, both, or neither!

For example, in any topological space, the entire set $X$ is open by definition. Its complement is the empty set $\emptyset$, which is also open by definition. This means $X$ is both open and closed. The same logic applies to the empty set.

Defining a topology by listing all its open sets can be cumbersome, especially for large sets. Instead, we can generate a topology from a smaller collection of sets called a basis.

A basis $\mathcal{B}$ for a topology is a collection of open sets such that every open set in the topology can be written as a union of sets from the basis. Think of it like a set of Lego bricks; you can build any required shape (any open set) by combining the basic bricks (the basis elements).

To be a valid basis, a collection of sets $\mathcal{B}$ must satisfy two conditions:

We can go one step further and use an even simpler collection called a subbasis. A subbasis $\mathcal{S}$ is any collection of subsets of $X$. We can form a basis from it by taking all possible finite intersections of sets in $\mathcal{S}$. Then, from that basis, we generate the full topology by taking all possible unions.

This two-step process—from subbasis to basis, then basis to topology—is a powerful way to construct complex topological spaces from simple starting points.

Let's look at a few ways to put a topology on a set $X$. For these examples, we'll use a simple set: $X = \{a, b, c\}$.

The trivial topology is the 'coarsest' possible, with the fewest open sets. It can't distinguish between any points. In contrast, the discrete topology is the 'finest,' with the most open sets. It isolates every point in its own open set.

These examples show that the same underlying set of points can have very different topological structures depending on which subsets we decide to call 'open'.

These foundational ideas—of defining spaces through collections of open sets—are the first step into the rich and fascinating field of topology.