Intuitive approach to topology

Question

I'm a highschool student (okay, almost a highschool student - it's summer) that's self-studying. I know some basics of naive set theory, linear algebra, and single-variable calculus. I'd like to give topology a shot. I looked up 'Topology without tears' and began reading.

It makes sense (so far, anyway) but there's not really any intuition on what exactly a topology is. It's all axioms and definitions, and while I know that's how most math books are written, I'd just thought I'd ask if there was a book that isn't written that way.

As this answer puts it:

You don't learn what a vector space is by swallowing a definition that says

A vector space $\langle V,S\rangle$ is a set $V$ and a field $S$ that satisfy the following 8 axioms: …

[...]

A good textbook will do this: it will reduce those 8 axioms to a brief statement of what the axioms are actually about, and provide a set of illuminating examples. In the case of the vector space, the brief statement I quoted, boldface in the original, was it: we can add any two vectors, and we can multiply vectors by scalars.

What's a textbook like that for topology? (I don't mind definitions; they're important. I'd just like some intuition to go with them.)

Answer 1

You might first want to study analysis, which will give you more of a motivation for learning topology. Analysis introduces you to many concepts in topology in a more tangible way, in more familiar contexts like the set of real numbers and metric spaces, where you at least have a notion of distance.

After analysis, you could study topology equipped with better intuition. This is the usual progression at the college level as well.

As for a text for introductory analysis, I recommended Principles of Mathematical Analysis by Walter Rudin. Chapter 2 covers the basic ideas of topology relevant to analysis.

Answer 2

The other answers are good, but appeal to ideas that might not be familiar to a high school student (metric spaces), or to things which I feel are an important consequence of topology but not a motivation/intuition (continuity, which requires extra ideas, like maps between topological spaces).

A topology is one of the weakest structures you can put on top of a bare set of objects. Very roughly, (my intuition is) it allows one to say which elements of the set are 'nearby' each other. It does so in rather an ingenious way - one which does not need to appeal to measurement or distance. Instead it uses the idea of open sets, which you may or may not be familiar with. One specifies 'a topology' by specifying its open sets, which are subsets with particular properties under unions and intersections. There are usually many possible distinct topologies available for any one set.

The open sets which contain a particular element from the big set are often called the 'neighbouroods' of that element. Sometimes one neighbourhood can lie entirely inside another (via set inclusion), so is a 'smaller' neighbourhood. Hence the intuition: one can think of the elements of a smaller neighbourhood as being 'closer' than those of a larger one.

This is then where we can talk about the ideas of metrics, which measure distance, or continuity, which requires one to talk about the relative 'sizes' of sets in the domain and image of a function. It's amazing that even with something so simple as a topology one can already talk about such powerful analytic ideas. But topologies on their own are so dreadfully unconstrained that some very strange structures are possible. Usually, at least in my line of work, a topology is used in conjunction with some other structure in which that intuition is realised in a precise sense. Then what the others have said - about continuity, etc - comes into play, where the topology does most of the low-level analytic heavy lifting.

Answer 3

We'd like to say:

A topological space is a set $X$ together with a notion of convergence relating sequences in $X$ with points of $X$. Given a sequence $x$ and a point $y$, we can ask whether or not $x$ converges to $y$, and certain axioms hold regarding this relation.

A topological space is called Hausdorff iff every sequence converges to at most one point.

A function $f : A \rightarrow B$ between topological spaces is continuous iff for all sequences $x$ in $A$ and all points $y \in A$, we have that if $x$ converges to $y$ in $A$, then $f(x)$ converges to $f(y)$ in $B$.

Unfortunately, this is completely wrong. For starters, you can't always recover the topology just from knowing which sequences converge to which points. Though sometimes you can; these are called sequential topological spaces. But in general, what your convergent sequence/point pairs isn't enough to determine the topology, and we have to pass to "generalized sequences"; the technical term is net. So, we want to say:

A topological space is a set $X$ together with a notion of convergence relating nets in $X$ with points of $X$. Given a net $x$ and a point $y$, we can ask whether or not $x$ converges to $y$, and certain axioms hold regarding this relation.

A topological space is called Hausdorff iff every net converges to at most one point.

A function $f : A \rightarrow B$ between topological spaces is continuous iff for all nets $x$ in $A$ and all points $y \in A$, we have that if $x$ converges to $y$ in $A$, then $f(x)$ converges to $f(y)$ in $B$.

This still doesn't quite work, because the "set" of all nets in a topological spaces turns out to be too big to form a set; they merely form a class. This creates some technical problems that we'd rather do without. The usual workaround goes like so:

• Given a set $X$, there's a notion of filter in a set.
• Every net corresponds to something called its "eventuality fiter."
• We can decide whether or not a net $x$ converges to a point $y$ just from knowing the eventuality filter of $x$.
• Hence instead of equipping $X$ with data regarding which net/point pairs are convergent, the usual workaround is to equip $X$ with data regarding which filter/point pairs are convergent, and treat convergence of nets as a derivative notion.

So, our definition becomes:

A topological space is a set $X$ together with a notion of convergence relating filters in $X$ with points of $X$. Given a filter $x$ and a point $y$, we can ask whether or not $x$ converges to $y$, and certain axioms hold regarding this relation.

A topological space is called Hausdorff iff every filter converges to at most one point.

A function $f : A \rightarrow B$ between topological spaces is continuous iff for all filters $x$ in $A$ and all points $y \in A$, we have that if $x$ converges to $y$ in $A$, then $f(x)$ converges to $f(y)$ in $B$.

There's still a subtle issue. If you try to axiomatize topological spaces via filter convergence, you wind up with a significantly more general notion called a convergence space. Convergence spaces are really nice, and personally I wish people would start treating them as the basic objects of interest in general topology, and treat topological spaces as a mere special case. Unfortunately, I haven't been able to find an elementary introduction to such things that I can link you to, so you'll have to learn things the classical way until you're ready to go off on your own.

Answer 4

One way to think about topology is that the elements of a topology are meant to be a generalization of the concept of an open set in analysis, without the need for a metric. The elements of a topology take some fundamental properties of an open set in analysis (that a countable union of open sets are open, and a finite intersection of open sets are open), and allows us to talk about open and closed sets without the need of a metric.

Thus under the definition of continuity, that the inverse image $f^{-1}(U)$ of an open set $U$ in the range of $f$ under some map $f$ is open in the domain of $f$, allows us to discuss continuity of maps between spaces that have no metric.

These are just some interesting ideas contained within basic topology - keep at it!

Answer 5

Topology is about defining the notion of continuous functions without ever talking about epsilons and deltas.

Answer 6

$\DeclareMathOperator{\int}{int}$I cannot post as a comment because of reputation but here is a related question on mathoverflow : https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/. I found the following intuition here.

We think of a region of the space as a statement which can be true or false depending on our location in the space (the statement being "we are in the given region").

We are given an unknown location (a point in the space). Among the true statements, we can prove some of them. The link with the geometric intuition is that a proof is an imprecise measurement of our location, like "$5 \pm 10^{-3}$". We cannot prove that a physical quantity is exactly $5$, but if it is indeed $5$, then we can prove that it is $5 \pm \varepsilon$ for all $\varepsilon > 0$.

The region/statement $V$ is a neighborhood of the location $x$ if we can prove $V$ when our location is $x$. The interior of $V$ is the set of points for which it is a neighborhood. The axioms are :

1. If we can prove something, then it is true : $\int(V) \subseteq V$.
2. If $A \implies B$ and if we can prove $A$, then we can prove $B$ : $V \subset W \implies \int(V) \subset \int(W)$.
3. If we can prove a finite number of properties, then we can prove their conjunction : $\cap_{i=1}^n \int(V_i) \subseteq \int(\cap_{i=1}^n V_i)$.
4. If we can prove something, then we can prove that we can prove it : $\int(\int(V)) = \int(V)$. (The space is well known, only the location in it is unknown.)

The first three axioms say that the neighborhoods of a point constitute a filter smaller than the principal filter at that point. The last axiom is the interesting one, it is a coherence condition between the different neighborhoods.

I find that this interpretation leads to interesting things.

Answer 7

I would recommend Elementary Topology. Textbook in Problems by O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev.

Answer 8

You should have a look into the following book: Topology by K. Jänich

The German version is a classic. Jänich is well-known for his excellent motivations; one just has to look at the section titles, for example "What is point-set topology about?," "What is algebraic topology?," "Homotopy - what for?", etc.

Furthermore, this book has more than 180 illustrations, so, yeah. At this point, I will let a reference on Amazon speak for me (as I can't formulate it better):

"It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to 'see into the future' of what kind of mathematics is waiting for them and how topology is indispensable in its study."

Answer 9

I'd suggest going in the opposite direction: Take a look at some unusual topological spaces (e.g., the Sorgenfrey line, the long line, the Zariski topology on a suitable space, etc.) to make it clear that topology isn't set up that way out of some sort of obstinacy or fetish for abstraction. I'd also recommend skipping ahead to some of the punchlines, even if you can't fill in the details yet: compactness, connectedness, etc.

To answer your specific question about a reference, I think there isn't a good one. Munkres' Topology is the canonical intro point-set topology textbook, but it's a tedious, pedantic slog. (I dislike Munkres' writing in general; his algebraic topology textbook, for example, sucks all the life out of a very interesting subject.) It might be useful to look at the appendix of Hatcher's Algebraic Topology on CW-complexes to see the sort of spaces that are often worked with in practice, and that are complicated enough to be interesting but still tractable.

Answer 10

Look at the half-open interval $(0,1] = \{ x : 0<x\le 1\}. $

Let the basic open sets in this set be open intervals, and half-open intervals with their right endpoint at $1$. (Thus $(0.6,0.8)$ is a basic open set and so is $(0.6,1],$ but $(0.6,0.8]$ is not.) Let the open sets be unions of basic open sets. (Unions of intervals -- either finitely many intervals or an infinite sequence of intervals.)

That's a topological space.

Now consider what happens if we alter our definition of "basic open set". Let us exclude half-open intervals ending at $1$ and instead let sets of the following form be included among the basic open sets: $$ (0, a) \cup (b,1]. $$ Thus every basic open set containing $1$ contains points at both ends of the interval.

That's a different topological space: topologically (but not metrically) it's a circle. The two ends of the interval are now glued together.

Answer 11

You said you know some basics of single-variable calculus. Definitions and standard proofs of the theorems of single-variable calculus use the $ε$-$δ$ formalism. If you interpret expressions of form $|x - y|$ as a distance between two points $x, y ∈ ℝ$, you may generalize many results from single-variable calculus to multivariable-calculus if you endow $ℝ^n$ with a distance. This leads to a natural notion of metric space: it is just a set of abstract points where you can measure distance between them (you have a function $d\colon X × X \to [0, ∞)$ satisfying natural axioms).

As others say, the more general notion of topology is enough to study continuity. But why, and what is continuity intuitively? I would say that a map $f\colon X \to Y$ is continuous if it maps “infinitely close things to infinitely close things”. Maybe we want to say that it maps “infinitely close points” to “infinitely close points”, but unless you formally introduce infinitesimals, this makes no sense because every two distinct points are not infinitely close (they have positive distance). But in a metric space you may also measure distances between points and sets – just put $d(x, A) := \inf\{d(x, y): y ∈ A\}$. And now it makes sense: a map between metric spaces is continuous if and only if for every $x$ and $A$ we have $d(x, A) = 0 \implies d(f(x), f[A]) = 0$.

Note that we don't need to know the exact distances between all points and sets, it is enough to know when the distance is zero, i.e. when a point is “infinitely close” to a set. And this is exactly the information stored in a topology. As an equivalent basic notion we may use the closure operator that assings to every set $A ⊆ X$ its closure $\overline{A}$ and that satisfies certain axioms. A map $f\colon X \to Y$ between topological spaces is continuous if and only if $x ∈ \overline{A} \implies f(x) ∈ \overline{f[A]}$, which is consistent with our interpretation “$x$ is infinitely close to $A$” and which generalizes $d(x, A) = 0$.

I think this gives some intuition even though it is expressed using the closure operator instead of open sets. But this doesn't matter since once you have any of open sets, closed sets, closure operator, interior operator, you have all of them. And realizing the connections between them is a basic exercise when introducing topological or even metric spaces.

Answer 12

You might want to try Basic Topology by M. A. Armstrong. I actually haven't read all of it nor taken a topology class, and I had some background in analysis when I read the first few chapters, but the book does try to motivate a lot of the concepts of topology.

For example, when introducing a topology, it does so by introducing an alternate definition of continuity. For any calculus student, a function $f$ is continuous at some point $a$ in its domain if and only if for each real $\epsilon > 0$, there exists a real $\delta > 0$ such that for each $x$ satisfying $|x - a| < \delta$, we have $|f(x) - f(a)| < \epsilon$. We say that $f$ is continuous if $f$ is continuous at each point on its domain.

Armstrong presents an alternate definition that, instead of using the $\epsilon$-$\delta$ definition, like the one above, rather uses neighborhoods, and proves this. After doing so, he describes "a topology on some set" as having the minimal requirements in order to define continuity of a function between two sets, assuming we use continuity with neighborhoods rather than using $\epsilon-\delta$. (Of course, much more detail goes into this in the actual book.)

The reason we want to make this abstraction is because the $\epsilon-\delta$ definition relies on a metric in order to work. A definition that relies on whether some set is a neighborhood of some point is a lot more general idea. Armstrong develops more stuff such as open sets, compactness, fundamental groups throughout the later chapters.

Personally, I thought the book was amazing at explaining the intuition behind topology, but again, I only read the first few chapters, so I have no idea how this book is overall. Maybe, the reviews on Amazon will help?

Answer 13

Often, familiar/intuitive implies more complex, and simpler implies unnatural/less obvious. Topology is a great illustration of this.

The definition of "topological space" is several times shorter, and its axioms are of a simpler logical form, than the definition of "vector space" + the definition of "field". Yet, it's easier to visualize and remember what happens in a linear transformation in the Cartesian plane (i.e. $\mathbb{R}^2$) than to visualize and remember what happens in a continuous function between topological spaces with some restriction or another.

This is because the Cartesian plane is specifically restricted so that it mimics a physical canvas that we could draw on. Even its esoteric aspects, like the completeness axiom, are carefully chosen to reduce the amount of gotchas and counterexamples that could cause an intuitive idea to be technically incorrect. You usually don't have to think about the axioms to solve a problem; you just use common sense.

Meanwhile, topological spaces begin with extremely few restrictions. General topology is an exercise in figuring out the bare minimum of assumptions you need to make in order to draw the conclusions you want to draw about a space or a function. So you'll see dozens of little restrictions that are defined abstrusely (but still much simpler than $\mathbb{R}^2$) and piled on top of topological spaces in a piecemeal fashion, all so that a theorem can say "if and only if" instead of just "if". And it often requires a clever, ad-hoc counterexample to show why a weaker assumption wouldn't be restrictive enough to imply what you want to imply.

In other words, topology questions all those assumptions you've taken for granted. To appreciate (and therefore remember) it, you have to take those assumptions for granted in the first place. That means, for example, proving theorems that rely on the behavior of familiar metric spaces (like $\mathbb{R}$ and $\mathbb{R}^\mathbb{R}$) when points get arbitrarily far away or arbitrarily close to each other. Or figuring out in what way you need to measure or approximate something to guarantee that it has stability within a desired tolerance.

So, for someone in the "curious about undergraduate-level math" stage of education, I'd recommend How to Think About Analysis. No, it's not a topology book. But it gets you thinking about what you need to think about in order to appreciate topology, and it discusses why each new thing is important before it enumerates the technical details of that thing.