Topology is often called "rubber-sheet geometry" because it studies properties of shapes that stay the same even when you stretch, bend, or twist them. To do this, we need a way to talk about nearness and connection without relying on fixed distances or angles. The fundamental structure that lets us do this is called a topological space.

A topological space is simply a set of points, which we'll call $X$, paired with a collection of its subsets, which we'll call $\tau$ (the Greek letter tau). The subsets in $\tau$ are special; we call them open sets. For $\tau$ to be a valid topology on $X$, it must follow three simple rules.

These rules provide a flexible way to define the "structure" of a space. The collection of open sets, $\tau$, tells us which points are "near" each other without using a ruler.

This definition might seem abstract, so let's look at a few examples on a simple set, $X = \{a, b, c\}$. Different collections of subsets can form different topologies on this same set.

The trivial and discrete topologies represent two extremes. The trivial topology is the "coarsest," lumping everything together. The discrete topology is the "finest," separating everything out. Most interesting topologies lie somewhere in between.

Another very important source of topologies comes from spaces where we can measure distance.

Any metric space automatically gives us a topology. The open sets are formed by taking unions of "open balls." An open ball centered at a point $p$ with radius $r$ is the set of all points whose distance from $p$ is less than $r$. This is the standard topology we intuitively use when thinking about 2D or 3D space.

The open sets are the stars of the show, but their counterparts, closed sets, are just as important. The definition is refreshingly direct.

The complement of a set $A$ within a larger set $X$ is everything in $X$ that is not in $A$. Let's go back to our earlier example where $X = \{a, b, c\}$ and the topology was $\tau = \{\emptyset, \{a\}, \{a, b\}, \{a, b, c\}\}$

• The set $\{a\}$ is open. Its complement is $\{b, c\}$, so $\{b, c\}$ is closed.
• The set $\{a, b\}$ is open. Its complement is $\{c\}$, so $\{c\}$ is closed.
• The empty set $\emptyset$ is open. Its complement is $X=\{a, b, c\}$, so $X$ is closed.
• The whole set $X$ is open. Its complement is $\emptyset$, so $\emptyset$ is closed.

Listing out every single open set can be tedious, especially for infinite sets. Often, it's easier to define a smaller collection of sets that can "generate" the entire topology. This is called a basis.

Once you have a basis, the full topology is defined simply: an open set is any set that can be written as a union of sets from the basis. Think of the basis sets as fundamental building blocks. You can create all the open sets you need just by combining these blocks.

For example, in the standard topology on the 2D plane, the collection of all open disks (the inside of circles) forms a basis. Any open set, no matter how weirdly shaped, can be described as a union of these simple open disks.

By choosing different bases, we can generate different topologies on the same underlying set of points, giving it entirely different geometric properties. This simple foundation of a set and its open subsets is the starting point for the entire field of topology.