How the Axioms and Intuition of a Topological Space Relate

I remember when moving into general topology and was given the definition of a topological space. In some sense, I had intuited they were encoding a general sense of ”openness” without relying on metrics, hence that they were more general than metric spaces. The various topological concepts like continuity and neighborhoods can be characterized by open sets, without the use of distance functions. This was the general intuition I had about topological spaces.

It can often be difficult to understand how a given set of axioms for an object even give rise to the intuition we have about it. For instance, think if a vector space, what is it fundamentally? A space with arrows that have both a direction and magnitude? Algebraically, they are sets equipped with a certain structure of linearity. That’s it. But it was only after abstract algebra (which I read after topology) I cultivated this way of thinking about mathematics in general.

Let's look at a topological space:

Let $X$ be a non-empty set and $\tau$ a collection of subsets of $X$ such that

1. $X\in \tau$,
2. $\emptyset \in \tau$,
3. If for each $\alpha \in I$, $O_\alpha \in \tau$, then $\bigcap _{\alpha \in I} O_\alpha \in \tau$, and
4. If for each $\alpha \in I$, $O_\alpha \in \mathcal J$, then $\bigcup _{\alpha \in I} O_\alpha \in \tau$.

The pair of objects $(X, \tau)$ is called a topological space. An open set is, by definition, an element of $\tau$.

This was at a point were I still unused to axiomatic constructions in math, simple assumptions that create expansive worlds. The hard part I remember was to connect my intuition to the axioms themselves. A topological space is a collection of objects or points, and a structure that that allows us to speak of nearby points or points that in some sense are close together. In open sets, we can always "move around freely" without hitting a boundary. So how does the four simple assumptions in the definition give this specific intuition?

So we have a set $X$, and a specific set $\tau$ of subsets of $X$. However, for the object $(X,\tau)$ to be a topological space, we must ensure that they follow Axioms 1-4:

Axiom 1

We require the whole space $X$ to be open because, intuitively, openness is meant to capture local properties and continuity over the entire space. By declaring $X$ as open, we assert that the entire set can be thought of as a region without any boundary issues within the space itself. Without $X$ being open, our framework would lack a fundamental "starting point," and many theorems in topology would not hold.

Axiom 2

The empty set is considered open because it acts as the zero element in our structure, analogous to the identity in algebraic structures like vector spaces and groups. In many mathematical settings, we include the trivial case (the empty set) to ensure that operations (like unions and intersections) behave well. For instance, the intersection of disjoint open sets should yield the “smallest” possible open set, and the empty set plays that role.

Axiom 3

The “overlap” of finitely many open sets should still be an open set. Intuitively, if you’re at a point in some space where several open regions meet, you expect that intersection to also be an open environment. If each neighborhood around a point is open, then any finite intersection of these neighborhoods should still allow you to “move around freely." Note that we only demand finite intersections here. Allowing arbitrary intersections would often force too many sets to be open (or not, depending on the topology), which would disrupt many natural examples. For instance, we could intersect an infinite number of open intervals $( -1/n, 1/n )$, which results in the set $\{0\}$. On the real line, a single point isn't open because you can't move around freely without hitting the "outside"

Axiom 4

You can “cover” parts of a space with open sets, and combining them results in another open set. Intuitively, you’re never “closing off” any part of the space by taking a union of open sets. No matter how many you take (even infinitely many), their union is still a space where you can locally move freely.

What are open sets here?

One day I remember just walking around a bit frustrating just asking ”what are open sets really then?” If open sets are defined in analysis using metrics, open balls, then what are open sets in topology? Just elements of $\tau$ such that these 4 axioms hold? It felt a bit amazing, but I want to argue that it’s a lot about acceptance. An open set is literally just an element of the $\tau$. That’s it. That’s a definition. But accepting these abstractions is much easier if truly see how their constitutive axioms connect to their intuition.