How to (Better) Understand Topology

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If you’ve only ever heard one thing about topology, it probably went something like this: “Topology is the subject in which donuts and coffee cups are considered identical, because one can be transformed into the other without cutting or gluing.” The idea is best communicated via gif:

On the other hand, if you’ve studied topology in depth, you know that the concept of “open set” is a core feature of the subject. Indeed, the way to define a particular topological space (say, the donut) is to begin with a set (specifically, the set of all points in the space) and then to identify precisely which subsets of those points will be considered “open.” Here’s an example for the donut (or torus, in proper math terminology):

In this diagram, the open set U is a “patch” along the surface, which doesn't include its (dashed) boundary. The collection of all open sets completely characterizes the space, meaning every property (as far as the subject of topology is concerned) is determined purely based on which sets are open.

At first glance, it isn’t clear how the donut–coffee cup transformation is related to the specification of open sets. Several questions arise:

• What’s the intuition behind these open sets?

• How do they encode information about the “shape” of a space?

• What key idea(s) is topology intended to capture, anyway?

• How do these ideas provide a framework for describing donut–coffee cup transformations?

In other words, what are the core concepts of topology, and how does the mathematical machinery of topology reflect these concepts? Compared to other areas of pure math (say, linear algebra or group theory), you could say that topology has a wide intuition–technicality gap:

The goal of this post is to help narrow the gap by extending both sides of the above canyon to bring them as close together, conceptually, as possible. The first key insight is this: It’s hard to understand the intuition behind open sets because they aren’t the right tool for the job. Open sets are a crucial tool for doing topology, but they aren’t the best tool for understanding it. There’s a better approach, using what are called the Kuratowski closure axioms, although to my knowledge they are rarely taught.

In Part 1 we’ll highlight some of the confusing aspects of the standard treatment, and in Part 2 we’ll explore the alternative approach and see how it elucidates the subject. By the end, our mental picture of topology will be better represented by the following diagram (whose details need not make sense yet).

Let’s bridge the gap.

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Part 1: The Problem

We wish to answer:

1. What is the intuition behind topology?

2. How is the intuitive side of topology is reflected by the technical side?

Before attacking these, let’s apply the same questions to a couple other subjects to get a sense of what a good answer should look like. To understand an area of pure math, we should be able to fill in the blanks of the following blurb:

In the subject of ___, we define a ___ as a set (whose elements we call ___s) with the following extra structures: ___, ___, ... . These structures provide the minimal possible framework to encode the concept(s) of ____. We tend to study the properties of ____s, which are functions that preserve the above structures.

I’ll start by filling in the blanks for linear algebra. (If you have ever studied linear algebra, I encourage you to scroll up and try completing the sentence yourself first.)

In the subject of linear algebra, we define a vector space as a set (whose elements we call vectors) with the following extra structures:

1) A binary operation +, which maps any pair of vectors to another vector, and

2) A binary operation ⋅, which maps any pair consisting of a real number and a vector to another vector.

These structures provide the minimal possible framework to encode the concept of linearity. We tend to study the properties of linear maps, which are functions that preserve the above structures.

Many details were omitted from this summary, such as the fact that vector spaces can be defined over more general number fields than just the real numbers, and that the operations + and ⋅ need to obey several axioms. But the blurb conveys at least a skeleton of the subject. Even if you have a minimal background in math, you might suspect that linearity is an important and foundational concept. You might sense that linearity has to do with how things scale ( ⋅ ) and involves quantities that can be added (+).

What does it mean for a function, f, to “preserve” these structures? For the + operation, it means that if A + B = C, then f(A) + f(B) = f(C). A similar statement holds for the ⋅ operation. You couldn’t really ask for a more reasonable formulation of what should constitute a “well-behaved function,” given the materials at hand.

Thus, I claim the intuition–technicality (IT) gap for linear algebra is not as wide as it is for topology. The intuitive side is the concept of linearity; the technical side involves a set of objects that can be added to each other and scaled by numbers, and it focuses on functions that preserve all the addition and multiplication structures. This outline is sparse, yet it successfully conveys the basic idea of an entire discipline. I’m not saying the subject is easy or simple by any means, but just that it’s possible to zoom all the way out and give a broad overview, as we have just done.

Let’s do one more, for group theory.

In the subject of group theory, we define a group as a set (whose elements we sometimes call actions) with the following extra structure:

A binary operation ⋅, which maps any pair of elements to another element and obeys the three group axioms.

This structure provides the minimal possible framework to encode the concept of symmetry. We tend to study the properties of homomorphisms, which are functions that preserve the above structure.

It doesn’t take too much further explanation to reveal the connection between the binary operation ⋅ and symmetric actions, though I won’t spend much time on this. Briefly, if you perform two symmetric actions consecutively on some geometric object, the end result is still a symmetric action. When the operation ⋅ is considered as a composition of two actions, then the group axioms (which I’ve omitted) start to feel well motivated and even natural.

In light of this, I claim group theory has a small IT gap, despite being a rich—and often difficult—subject with far-reaching results.

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Let’s now try filling in the blanks for topology. Beware, though, that this first attempt will turn out to be problematic, and we’ll need to modify it for reasons that will be addressed soon.

In the subject of topology, we define a topological space as a set (whose elements are called points) with the following extra structure:

A collection of distinguished subsets of points, called “open sets,” which obey the three open-set axioms.

This structure provides the minimal possible framework to encode the concept of limit. We tend to study the properties of continuous maps, which are functions that preserve the above structure.

There are two problems with this description. First, there is no clear intuition behind behind what the open sets are supposed to represent, or how they encode the concepts of limit or continuity. To be sure, there is indeed a way to define limits in terms of open sets, which first-time topology students learn early on. There must be a way, since limits are one of the core features of topology, and the standard “open set” approach is by no means broken. It’s just that the open-set approach doesn’t lend itself to the clearest possible picture of what limits are, how they’re encoded in the technical apparatus, and why exactly continuous functions end up preserving them.

Second, the final sentence of the blurb is flat-out wrong, if interpreted in the same way as the the previous examples. Continuous functions do not preserve the structure of open sets, because they can map open sets to non-open sets. There is a sense in which continuous functions preserve open sets, but it works in the opposite direction from usual. Hence, first-time topology students must learn the continuous-function mantra, “Pre-images of open sets are open.” We are left with the question of why (on an intuitive level) the interesting maps of topology preserve the defining structure in the reverse direction, while in other areas of math, we mainly consider the forward direction. As far as I can tell, this question has no fully satisfying answer, except perhaps for the argument that open sets should not be considered the “defining structure” of topology after all.

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The general sense of mystery surrounding these issues is by no means restricted to first-time learners. On the contrary, I think many people who are familiar with topology, and even some experts, would struggle to address these questions. Consider the opening paragraph of this question posed by a university professor on MathOverflow, a Q&A website used mainly by professional mathematicians:

I’m ashamed to admit it, but I don’t think I’ve ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of many examples, but it’s never been obvious to me how it came about, compared, for example, to the rather intuitive definition of a metric space. In some ways, the sparseness of the definition is startling as it tries to capture, apparently successfully, the barest notion of “space” imaginable.

– Minhyong Kim

The question has been viewed over 77 thousand times and has received two pages–worth of widely varying answers given by other experts, many of which are followed by long debates in the comments. (I have never seen the answers to any other post reach two pages!) Even among experts, evidently, the intuition behind topology is a hot topic with numerous competing opinions and no authoritative final word.

The approach I’ll follow in the next section is related to one of the answers from that post, and it is by no means a new or original idea. It was proposed by the mathematician Kazimierz Kuratowski in 1920, and it has since been occasionally promoted in academic journals and mathematics education resources for its conceptual and pedagogical advantages. This is not even the only blog post about the topic, although I have not yet seen any neatly packaged explainers for first-time topology students and other math enthusiasts. Nor have I seen this approach taken in any textbooks. After 100 years, it’s time to spread Kuratowski’s insight to a wider audience.

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Part 2: A Solution

Our first stab at a conceptual overview of topology was at least on the right track in highlighting limits and continuity as central themes. Let’s take a step back and consider limits in a more concrete setting. In particular, let’s examine a sequence of real numbers:

Even with no prior training, you might guess (correctly) that the limit of this sequence is zero. In a real analysis course, one learns how to justify this intuition with an exact definition and proof. But the following explanation gets the idea across: Even though no entry of the sequence is ever equal to zero, the set of all numbers in the sequence gets “infinitely close” to zero. (The more accurate terminology is “arbitrarily close,” but I think “infinitely close” is more colorful and not too far off base.)

You should imagine the infinitely many numbers in this sequence as points clustering right up against zero, without actually falling on top of it.

As another example of infinite closeness, consider the “soft” unit disk, D, which is the set of all points in the plane whose distance from the origin is less than (but not equal to) one:

The boundary is dashed to indicate that its points are not included in the set D. However, each point along the boundary is infinitely close to D. In an analysis course, this statement would be made more precise, but hopefully the idea feels reasonable enough, even if you have not taken such a course.

From now on, let’s say that if a set of points S is infinitely close to some point p, then

“S contacts p.”

For example, if we let S denote the set of all numbers in the above sequence, then we can say “S contacts zero.” Or in the second example, if p is any point on the boundary of D, then D contacts p.

Also, if a point p is in a set S, then S contacts p; it seems reasonable enough that any set is infinitely close to each of its own points. The following diagram summarizes the intuition we’re trying to build for the “contact” relation:

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The examples discussed so far have been situated in a special kind of space called a metric space, in which there is a natural way to determine the distance between any two points. I didn’t give the technical definition of what infinite closeness means in this context, because I could rely on your spatial intuition to give you a feel for the concept.

But topology sets its sights higher than these familiar spaces. In fact, one of the broad goals of topology is to remove the notion of distance from geometry and see what remains. Kuratowski’s idea was that, in order to define a ”space” in the most general setting possible (where even a notion of distance might not be available), it suffices to specify all of the contact relations between sets and points. You could say that we only need to keep the weakest possible notion of distance, in which a point can either be infinitely close to a set, or not, with no further detail.

Given a set of points that you’d like to turn into a topological space (say, the set of all points in the donut/torus), to define a topology on the set is to specify which points are contacted by which subsets—no more and no less. In other words, the information defining any particular topological space is contained in a function that, given any set of points A in the space, returns the set of all points contacted by A. This set is called the closure of A, and is denoted Ā; the function, which takes each set to its closure, is called the closure operator.

The closure operator and the “contact” relation are equivalent structures. Since Ā is the set of all points contacted by A, the statement “A contacts p” is equivalent to the statement “p is a member of Ā” (symbolically, p ∈ Ā):

There are also four axioms governing the closure operator, which we’ll look at soon. But first let’s take a moment to appreciate some benefits of defining a topological space in this way.

First, we now have a direct connection between the core ideas of topology (infinite closeness and limits) and the technical apparatus (the closure operator). The closure operator is “designed” to specify which points are infinitely close to any subset, and infinite closeness is the key intuition behind limits. There are still some subtleties we would have to address before arriving at the precise definition of, say, the limit of a sequence; but starting with the closure operator, we can feel confident that all the necessary materials for such a definition are at hand.

Second, continuity becomes completely natural in this context. We have defined topology in terms of a fundamental relation “A contacts p,” and this relation is precisely the structure that continuous functions preserve. Continuous maps can now be defined in close analogy with the “special” maps of linear algebra and group theory, as indicated by the following comparison chart.

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Subject f is a ... means ...

Linear algebra: linear map A + B = C ⇒ f(A) + f(B) = f(C)

t⋅A = C ⇒ t⋅f(A) = f(C)

Group theory: homomorphism x⋅y = z ⇒ f(x)⋅f(y) = f(z)

Topology: continuous map S contacts p ⇒ f(S) contacts f(p)

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In each case, the well-behaved functions are the ones for which we can simply “apply the f” to each object in the simplest possible statements that the subject allows us to assert.

The function transforming a donut into a coffee cup, mentioned at the start of this post, is continuous. This approach to topology paints a clear mental picture of what that means:

Since S contacts x in the above picture, f(S) must contact f(x). The fact that this preservation holds for any subset S and point x is what makes f continuous. By the way, the donut-coffee cup transformation is actually even better than continuous: It is invertible (the process can be run backward), and the reverse transformation is also continuous. Such a function is called homeomorphism, and when one exists between two spaces, it means the topological properties of both spaces must be identical.

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Yet another advantage of Kuratowski’s approach is that its axioms (which I have so far ignored) also have an intuitive appeal, unlike those of the traditional open-set formulation. To define a topology is to specify a closure operator, but not just any such operator will do: A valid closure operator must obey the four closure axioms, which we’ll briefly examine.

The first axiom concerns the empty set, ∅:

That is, the empty set is not infinitely close to any points—seems fair enough.

The second one is a restatement of an idea we’ve already touched on:

That is, any set A is contained in its closure. This is equivalent to the reasonable assertion that any set A contacts all of its own points. The closure operator can only extend a set, not reduce it.

The third axiom is more subtle:

The left side denotes the closure of the closure of A, so the axiom asserts that applying the closure operator twice is the same as applying it once. This property can be motivated from an infinite-closeness perspective: If you take all the points infinitely close to A and tack them onto A, the new set (namely, Ā) cannot be infinitely close to any “new” points. Loosely speaking, we could say that infinite closeness is transitive: If A is infinitely close to all the points in Ā, and Ā is infinitely close to some point p, then A must have already been infinitely close to p.

Finally, we have

That is, the closure of the union of A and B is the union of their closures. Equivalently, if the union of A and B contacts p, then either A or B must contact p. Intuitively, A and B combined only have the power to contact points that at least one of them already contacted. Taking the union of two sets cannot magically extend their reach.

The last two axioms admittedly might feel less “inherently necessary” than the first two, but there is at least a mental picture associated with each of them that relates directly to the concepts of infinite closeness and limit. On the other hand, the three open-set axioms of the standard approach are notoriously difficult to motivate (although some answers on the aforementioned MathOverflow post attempt this task).

Both sets of axioms appear on the “technicality” side of the second IT gap / canyon diagram. The equivalence arrow ⇔ between them is not just metaphor. Given a topology defined using either structure (open sets or a closure operator), one can prove that the other structure arises automatically and obeys its corresponding set of axioms. Proving this equivalence is a worthwhile exercise for any topology student; if you’re in that category, I recommend trying it yourself first, and referring to the resources in the previous links when you get stuck.

This equivalence highlights the fact that Kuratowski’s approach provides new understanding, but not in principle any new math. Open sets should not (and probably could not) be banished from the subject—they are essential when doing topology in practice (that is, when defining topological properties and proving theorems). But the intuition to be gained from the closure-operator formulation is valuable in its own right, and it even seems to have led Kuratowski to discover some new results that became more natural to consider and explore under his framework. In fact, research is still being done on generalizations of his original theorems.

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I’ll end by completing the topology blurb once more, this time the right way:

In the subject of topology, we define a topological space as a set (whose elements are called points) with the following extra structures:

1) A binary relation called “contact” between sets of points and individual points, or equivalently...

2) A map called the closure operator, which maps any set of points A to the set of all points contacted by A.

This structure provides the minimal possible framework to encode the concepts of “infinite closeness” and limit. We tend to study the properties of continuous maps, which are functions that preserve the above structures.