# An Introduction to Topology

When I took a poll of topics that people wanted my to write about, an awful lot of you asked me to write about topology. I did that before – right after I moved my blog to ScienceBlogs. But it’s been a while. So I’m going to go back to those old posts, do some editing and polishing, correct some errors, and repost them. Along the way, I’ll add a few new posts.

I’ve said before that the way that I view math is that it’s fundamentally about abstraction. Math is taking complex ideas, breaking down to simple concepts, and then exploring what those concepts really mean, and exactly what you can build using them.

I argue that topology, at its deepest level, is about continuity and nearness. In a continuous surface, what does in mean for things to be close to one another? What kind of structures can you build using nothing but nearness relationships? What is a structure defined solely by that notion of nearness?

When I say that continuity is fundamental to the basic abstraction of topoligy, some people – particularly topologists! – get upset, because continuity has a very specific meaning within topology, and that is not the meaning that I’m using when I make that statement.

In topology, you take a set of points, and you define their structure by which points are close to each other. Most of the basic topological structures are built using infinite sets of points, where there is no real notion of two points that are immediate neighbors; there’s a continuum of ever smaller regions that define ever closer and closer neighborhood relationships. The kind of continuity that I’m talking about is more like the continuity of a surface – it’s that fundamental notion of things that are close to each other, with that infinite ability to get take narrower and narrower subsets around a point. The kinds of structures that you get from doing that are interesting. In some sense, you’re defining shapes – but they’re malleable, squishy, twisty shapes, because all that matters is what points are close to what other points – not what direction you need to go to get from one to another.

Let’s take a quick look at an example. There’s a famous joke about topologists; you can always recognize a topology at breakfast, because they’re the people who can’t tell the difference between their coffee mug and their donut.

Like most good jokes, there’s a kernel of truth hidden inside of it. From the viewpoint of topology, the coffee mug and the donut are the same shape. They’re both toruses. In topology, the exact shape doesn’t matter: what matters is the basic continuities of the surface: what is connected to what, what points are close to what other points. In the following diagram, all three shapes are topologically identical: If you turn the coffee mug into clay, you can remold it from mug-shape to donut-shape without tearing it, or breaking it, or gluing any edges together. One can become the other by just squishing and stretching: so in topology, they are the same shape.

On the other hand, a sphere is different: you can’t turn a donut into a sphere without punching a hole in it; and you can’t turn a sphere into a torus without either punching a hole in it, or stretching it into a tube and gluing the ends together. You can’t turn one into the other without changing the basic continuity of the shape.

To look at it slightly more formally: take a sphere. Now, punch a hole through it, to turn in into a torus. If you think about the points that surround the donut-hole, they used to be close – that is, neighbors of – the points on the other side of the hole. But after the hole is punched through, they’re really far away – you need to go all the way around the hole to get to them, when they used to be right next door. So you’ve changed the nearness relations by making that hole.

Topology was one of the hottest mathematical topics of the 20th century, and as a result, it naturally has a lot of subfields. A few examples include:

Metric topology
the study of distance in different spaces. The measure of distance and related concepts like angles in different topologies.
Algebraic topology
the study of topologies using the tools of abstract algebra. In particular, studies of things like how to construct a complex space from simpler ones. Category theory is largely based on concepts that originated in algebraic topology.
Geometric topology
the study of manifolds and their embeddings. In general, geometric topology looks at lower-dimensional structures, most either two or three dimensional. (A manifold is an abstract space where every point is in a region that appears to be euclidean if you only look at the local neighborhood. But on a larger scale, the euclidean properties may disappear.)
Network topology
topology in the realm of discrete math. Network topologies are graphs (in the graph theory sense) consisting of nodes and edges.
Differential topology
the study of differential equations in topological spaces that have the properties necessary to make calculus work.

Personally, I find metric topology to be remarkably dull, so I’m not going to write much about it. Differential topology is completely beyond the capabilities of my puny brain, so there’s no way that I can write anything intelligent about it. Network topology more properly belongs in a discussion of graph theory, which is something I’ve written about before. So I’ll give you a passing glance at metric topology to see what it’s all about, and algebraic topology is where I’ll spend most of my time.

## 52 thoughts on “An Introduction to Topology”

1. Rob T.

Topology always makes me think of the short stories in Fantasia Mathematica.

2. Sharkey

Nice, looking forward to hearing more about topology, even if I suggested another topic back on SB 🙂

I’ve been reading Topology without Tears (http://uob-community.ballarat.edu.au/~smorris/topology.htm), and so far it seems like a good intro to the subject. Just a quick question: will you be discussing topoi when you get to the categorical stuff, Mark?

3. John Armstrong

I think there’s a typo:

I find metric topology to be remarkably dull

I’ll give you a passing glance at metric topology

I take it one of these is supposed to be “geometric”? Which one?

Incidentally.. what you’ve described as “geometric topology” isn’t really any different from differential topology. What do you think the difference is?

1. MarkCC

No, not a typo.

I don’t like metric topology, but I also find that the very basic ideas of neighborhood are easiest to understand starting off with metric spaces. So there’s a bit of metric topology to get things started, building from metric spaces into topological spaces that have metrics. Not much, but some. THat’s why I said “a passing glance”.

In terms of geometric topology vs differential topology- they’re obviously closely related. The difference between them is exactly the same as the difference between plane geometry and plane calculus.

1. John Armstrong

And what is “plane calculus”?

I’m serious.. A Ph.D. in math, specializing in knot theory, and I haven’t seen the subject partitioned like this.

1. Sean

I think some of these terms may be more historical, and aren’t really used much anymore within the mathematical community (at least I haven’t heard them). In particular, I’ve never heard anyone use the terms geometric topology or differential topology.

Nowadays one of the hottest research fields in math is so-called “low dimensional topology” (almost everyone has at least heard of the Poincare conjecture).

1. John Armstrong

Differential topology still shows up a lot. I took my first class on it with Novikov in 1999.

Basically, it’s the notion of a “manifold”, which is locally-homeomorphic to n-dimensional real space, without the notion of a “metric” (not the same metric as in what Mark’s calling “metric topology”) that’s important in differential geometry. There’s a lot you can say topologically about manifolds before measuring them.

2. Doug Spoonwood

The notion of a metric space got generalized to that of a probabilisitic metric space. Probabilistic metric spaces get developed by refering to a so-called “triangle function” or “T-norm” T which satisfy these axioms (1 doesn’t necessarily mean the number 1, though it can):
T(a, b)=T(b, a),
T(a, T(b, c))=T(T(a, b), c)
T(a, 1)=a
If a <= b, then T(a, c) <= T(b, c).

These axioms also give us the structure of what almost always (and with very good reason also) gets used to model the notion of intersection in fuzzy logic and fuzzy subset theory (well the membership functions involved, elements combine in same way as they do in traditional set theory). The usual way of modeling union in fuzzy logic and fuzzy subset theory once replacing T by another function S, only requires one axiom to change.

S(a, b)=S(b, a)
S(a, S(b, c))=S(S(a, b), c)
S(a, 0)=a
If a <= b, then S(a, c) <= S(b, c).

If a, b, and c only take on values in {0, 1}, and T gets interpreted as the "meet" operator of a Boolean algebra and S gets interpreted as the "join" operator of a Boolean algebra, one can verify that (T, {0, 1}) with the above axioms also satisfy the axioms of a Boolean algebra.

Maybe you already knew all that though.

2. MarkCC

Sorry if I’m getting the name wrong: what I mean is simple two-dimensional calculus.

Back when I learned calculus, we started with calculus in a two-dimensional place, and learned to do basic differentiation and integration within the plane. From there, we moved on to multivariable calculus, where we could work in more dimensions. To me, it always seemed to be exactly the same distinction we’d made in geometry, where we started off entirely within a plane, and then took the basic ideas of doing geometry in that plane, and expanded them into more dimensions. Frankly, it never occurred to me that maybe I was creating that distinction myself; just seemed natural to call it plane calculus. What’s the proper name of basic two-variable calculus?

1. John Armstrong

The basic first course? I’d call that single-variable calculus. The fact that the graphs lie in the Cartesian plane is sort of incidental (although invaluable for visual intuition).

As for “geometric topology”, it seems sort of close to what people call “low-dimensional topology”. But that’s all just part of differential or algebraic topology. The reason we single out 2-D, 3-D, and sometimes 4-D examples isn’t just that those are what we can understand, but because there are a lot of special things that happen in some of these low dimensions.

For example, look at the Geometrization conjecture. Understanding all possible metrics (differential geometry) on 2-dimensional manifolds (differential topology) was essentially done by Gauss. Cataloguing all possible metrics on 4- and higher-dimensional manifolds is unsolvable, but what can happen is basically understood. But understanding all possible metrics on 3-dimensional manifolds is hard.

Similarly, consider “exotic R4”. It’s possible to have a metric on four-dimensional real space which is not equivalent to the standard one, and this is only possible in four dimensions. That is, if you show me a Riemannian n-manifold and prove to me that it’s homeomorphic to Rn, then I know that it’s also diffeomorphic to Rn if n≠4, but I don’t know this if n=4.

For knot theory, the question of which Sn an Sm can nontrivially knot in is amazingly detailed for low dimensions, and becomes more regular in higher dimensions.

2. Sean

John, exotic R^4 is a smooth structure that is not diffeomorphic to the normal smooth structure on R^4. Certainly there are Riemannian manifolds in any dimension n>1 that are homeomorphic to R^n but not isomorphic (eg. anything with nonzero curvature).

3. John Armstrong

You might think so, Sean, but no. There are no exotic R^n (evidently <sup&rt; tags don’t work here) except for R^4.

4. Sean

I know that there are no exotic R^n besides R^4- I was just nit-picking what you wrote. You say:

“It’s possible to have a metric on four-dimensional real space which is not equivalent to the standard one, and this is only possible in four dimensions.”

Here you’re talking about a metric, which I take to mean Riemannian metric, and “equivalent” presumably means isomorphic in the context of metrics. With this interpretation your statement is incorrect- which is what I pointed out. Of course I assume you meant to say,

“It’s possible to have a *smooth structure* on four-dimensional real space which is not equivalent to the standard one, and this is only possible in four dimensions.”

5. Doug Spoonwood

If this doesn’t come as clear already, John calls such single-varaible calculus, because the entire subject covers functions which have a single variable as their domain. Mark calls such two-variable calculus since given that we deal only with real numbers, both x and f(x) can get thought of as variable quantities which take on numbers as specific points.

A study of single-variable calculus can start with sequences (functions from ordinals to the real numbers). Each term of a sequence can get thought as a displacement of a number from an origin point on a real number line (which implies each term of the sequence can get represented by a displacement from the previous term), so from Mark’s geometric perspective one could start with 1-dimensional calculus.

Since series really just consist of a function over a sequence (at least where the function associates), and their partial sums, or products, or such can get represented accurately as displacements on a real number line one can also represent partial sums, or products, or such as discrete points on a real number line… in 1-dimensional space. Since both sequences and partial sums qualify as functions from the ordinals to the real numbers, one can also represent them 2-dimensionally. What does all of this imply?

6. Doug Spoonwood

Whoops, I forgot about constant sequences and sequences that repeat terms. A 1-dimensional representation of a sequence doesn’t always give us as much information as algebra does. But, if no terms ever repeat in the sequence, then a 1-dimensional representation makes sense.

2. Raskolnikov

Metric topology is also important in the study of topological vector spaces, since vector spaces are often naturally endowed with a metric through the scalar product (Hilbert spaces) or the norm (Banach spaces).

4. Pseudonym

I forget who it was (Dan Piponi?) who pointed out that topology has the greatest disconnect between what you think it’s about, and what it’s actually about. You think it’s going to be all cool stuff like toruses and Klein bottles, and you turn up to the first lecture to discover lots of material about open and closed sets.

1. Raskolnikov

That’s cause you only went to the introductory course of basic topology. The one that is given prior to most analysis (advanced calculus) courses. If you want Klein bottles and the likes of those, you should go to algebraic or geometric topology.

5. Tinyboss

I’m not sure that plain old topology has anything at all to do with “nearness”. In the absence of a metric, how do you say whether point x is “near” to point y?

1. Carsten

You can do almost everything in a topological space, that you can do in a metric space.
Nearness is not a great example. You can’t define it in a metric space either (is a point A “near” point B, when their distant is smaller than 1/2? Or 0.0001?) but essentially you abstract from the distant to “neighbourhoods” (you may think epsilon-balls in the metric).

A better example are limits of a series of points in the space.
You know the epsilon-delta-definition in a metric space?
Good – now you will normaly know the equivalent notion with epsilon-balls (let (a_n) be a series of points in a metric space with distance-function d, then you can define a e-Ball to a point A as the set of all points x with d(x,A) 0 you can find an integer N so that every a_n (with n > N) lies within the e-Ball to L)
This is exactly how you do it in an topology – but instead of the e-Balls you use the notion of open sets containing a point or a neighbourhood of this point:
L ist the limit of (a_n) if for every neighbourhood U of L there is an integer N so that every a_n (with n > N) lies in U.

1. Tinyboss

Of course you can define nearness in a metric space, at least as well as you can define it anywhere else, for instance in the physical world. Nearness is relative, but you can always say whether one distance is shorter than another. But without a metric, all you have to work with are open sets. I’m asking, how can you talk about “nearness” in that setting?

2. John Armstrong

With a metric — a notion of distance — “near” means “within a ball of some fixed (usually small) radius”. That’s an example of an open set, and the general properties of “open sets” were abstracted from that.

1. Tinyboss

Right, but to my understanding that has everything to do with continuity, and very little to do with nearness. With a metric, you can say with certainty whether y is nearer to x than z is to x (which is really all you can say in the physical world too, of course), but (as I understand it) you can’t in general say any analogous thing in a general topological space. Other than actual containment, there’s not really any way to say that a particular neighborhood of x is smaller than any other one, right?

1. Tinyboss

(Preface: John, thanks for taking the time to discuss this with me. I just want to say that any assertions I make are meant to describe my own understanding of the topic, certainly not to instruct.)

If I had to describe continuity without recourse to mathematical formalism, I think I’d sooner talk about connectedness than nearness. The statement some of us heard in pre-calc that continuous functions are “ones we can draw without lifting the pencil” certainly misses some important details, but it does a good job of intuitively conveying the fact that a continuous image of a connected set remains connected, and also the fact that you might move your pencil arbitrarily far, and that’s okay.

Topology doesn’t tell us how to turn a coffee mug into a donut–it tells us they are already the same thing. To me, that suggests that any idea of “nearness” that is meaningful in the general topological sense doesn’t correspond enough to the everyday idea of distance to merit the term.

2. John Armstrong

Yes, connectedness is preserved by continuity. But that isn’t the essence of continuity.

Continuity essentially comes down to the epsilon-delta definition, which covers it for metric spaces. However, not all spaces are metrizable, and they’re not all pathological. For one thing, algebraic geometry gives us perfectly reasonable spaces that we want to treat topologically, but which can’t be brought under the rubric of metric spaces.

When we start to ask ourselves, what properties of open metric balls are really important for continuity and the consequences of continuity we care about, we’re led to the notion of an open neighborhood, and from there to the generic definition of a topological space. In this generality, we say that a point y is “near” a point x if y is in some (usually “small”) open neighborhood of x.

3. Raskolnikov

“Not all open sets look like open balls, and not all nested sequences of open sets ever get “small”. ”

It’s true that choosing other open sets, it is possible to reverse the relation of nearness. I have overlooked that fact. But that is just because I have stipulated arbitrary open sets in my definition. If I were to use a basis of open sets, I guess I could make get rid of this arbitrariness. Note that this isn’t any weirder than changing metric. A point y could be nearer to x than z according to one metic but farther accoring to another.

1. Raskolnikov

What about saying that y is contained in more open sets containing x than z? Of course, you would need an appropriate counting measure if there is a continuous amount of open sets possible. Which would bring some measure theory into topology! Great! 😀

1. Raskolnikov

But basically, it is as John says. The notion of nearness is incorporated into the notion of continuity. When you can find a sequence of open sets \$O_n (n=0,1,2,…)\$ about \$x\$ such that \$O_n in O_{n-1}\$ for every \$n\$. Then saying \$y\$ is nearer to \$x\$ than \$z\$ is saying that \$z\$ is contained in all open sets up to \$O_p\$ for instance, while \$y\$ is contained in all open sets up to \$O_q\$ with \$q>p\$.

No metric needed. (And neither is a measure needed.)

2. Tinyboss

Take the plane R^2 in the standard topology (pretending we don’t have a metric). It’s clear that for any three distinct points x,y,z, you can find sequences of nested sets as you describe to “show” that x is nearer to y than z, and also that y is nearer than x to z.

Not all open sets look like open balls, and not all nested sequences of open sets ever get “small”. I still don’t think that topology alone ever gets us anything that can properly be called “nearness”.

3. Tinyboss

Sorry, it should read “x is nearer than y to z, and y is nearer than x to z”.

4. Raskolnikov

“Not all open sets look like open balls, and not all nested sequences of open sets ever get “small”. ”

It’s true that choosing other open sets, it is possible to reverse the relation of nearness. I have overlooked that fact. But that is just because I have stipulated arbitrary open sets in my definition. If I were to use a basis of open sets, I guess I could make get rid of this arbitrariness. Note that this isn’t any weirder than changing metric. A point y could be nearer to x than z according to one metic but farther accoring to another.

5. John Armstrong

But how is “near” any less arbitrary a term in a metric space, really? Is a distance less than 1 “near”? less than 0.1? less than 0.01?

Really even in a metric space we don’t say formally that a point y is “near” a point x. We say that it’s “within ε of x” — meaning it’s within an open metric ball of radius ε centered at x”. When we don’t have a metric handy, we say that it’s “within a neighborhood N of x”, where N is some open set containing x.

And the point is how we use them. For functions between metric spaces X and Y, we’re used to the ε-δ condition for continuity: for every ε there is a δ so that ρ_X(x,x_0)<δ implies ρ_Y(f(x),f(x_0))<ε. Colloquially, we say that if we restrict x to be "near enough" to x_0, we can make its image f(x) "arbitrarily near" to f(x_0).

But if we don't have open balls, we use open neighborhoods in place: for every open neighborhood M of f(x_0) there is an open neighborhood N so that if x is in N then f(x) is in M. That is again, if we restrict x to be "near enough" to x_0, we can make its image f(x) "arbitrarily near" to f(x_0). This is the definition for continuity given a "neighborhood base", which is equivalent to the usual one about preimages of open sets being open.

6. John Armstrong

But how is “near” any less arbitrary a term in a metric space, really? Is a distance less than 1 “near”? less than 0.1? less than 0.01?

Really even in a metric space we don’t say formally that a point y is “near” a point x. We say that it’s “within ε of x” — meaning it’s within an open metric ball of radius ε centered at x”. When we don’t have a metric handy, we say that it’s “within a neighborhood N of x”, where N is some open set containing x.

And the point is how we use them. For functions between metric spaces X and Y, we’re used to the ε-δ condition for continuity: for every ε there is a δ so that ρ_X(x,x_0)<δ implies ρ_Y(f(x),f(x_0))<ε. Colloquially, we say that if we restrict x to be "near enough" to x_0, we can make its image f(x) "arbitrarily near" to f(x_0).

But if we don't have open balls, we use open neighborhoods in place: for every open neighborhood M of f(x_0) there is an open neighborhood N so that if x is in N then f(x) is in M. That is again, if we restrict x to be "near enough" to x_0, we can make its image f(x) "arbitrarily near" to f(x_0). This is the definition for continuity given a "neighborhood base", which is equivalent to the usual one about preimages of open sets being open.

7. Tinyboss

@John, I can explain precisely what I mean when I say that “near” is less arbitrary in a metric space than in a general topological space (and have explained it, although it’s scattered across several replies).

In a metric space (or the everyday physical world for that matter), we don’t have an absolute notion of what counts as “near”, any more than what counts as “big” or “hot”. But we can always compare distances and say whether one distance is larger than the other. Hence, for any distinct points X, Y, Z, it is unambiguous whether X or Y is nearer to Z. And that’s the essential thing about “nearness” that is not captured, IMO, by a general topological space.

8. John Armstrong

You’re focusing on individual points instead of regions, and individual points tell you nothing about continuity. What a metric tells you is how to compare sizes of metric balls, but that comes down to containment, just as open sets do.

6. Susan B.

Lately I’ve been thinking that the “metric” version of continuity is VERY important to understanding the impact of topology, precisely because it is the aspect of continuity that topology does away with. That is, topology tells us very important things about metric spaces by telling us which properties do not depend on the metric. (Of course, there are plenty of topological spaces that are non-metrizable and so on, but that’s not my point.)

7. Carsten

Well,

of course topology has something to do with “continuity” – it’s the same situation that you have in group-theory with homomorphisms (or almost – the inverse of an bijective continious map between topologies don’t have to be continous) – category theory FTW

8. mattiast

Does “Network topology” really have anything to do with topology other than the name?

9. Tualha

If you turn the coffee mug into clay…

I’m pretty sure most coffee mugs are clay. Not in the sense you meant, of course.

10. --bill

Where does point-set topology fall in this classification?

1. MarkCC

I’m not sure; I haven’t studied point-set topology. My (limited) understanding is that it would probably fall under algebraic topology. Most of the other fields rely on manifolds and things that are similar to manifolds. Point-sets tend to have much more erratic structure, so a lot of the stuff that you do in typical topology don’t make sense.

1. John Armstrong

I’d say it goes something like this:

Topology in general is point-set topology

Metric topology is a part of that, restricting to spaces which are “metrizable”. In fact, a large portion of point-set topology is given to determining exactly which topological spaces have a compatible metric.

Differential topology studies those spaces which are locally isomorphic to topological vector spaces (like Euclidean space). These are almost always metrizable.

Algebraic topology is the use of abstract algebraic techniques to study topological spaces. This is usually done with the use of functors from a nice subcategory of topological spaces to some category of algebraic objects. The need to restrict to a “nice” subcategory usually translates to the use of CW-complexes, which are slightly more general than manifolds, and are again usually metrizable.

2. Sean

Point-set topology is what you call topology when you are teaching a real-analysis course 😉

Seriously though, I think people normally use the term point-set topology to refer to theorems about topology that don’t have anything to do with manifolds (or perhaps aren’t motivated in some way by the study of manifolds). For example the type of topological vector spaces used in Functional analysis, or the Baire category theorem and it’s various consequences. In this type of setting you normally have a space of functions in mind.

11. Bard

Point-set topology too! My favorite!

1. MarkCC

Hey, Bard, great to see you here! (for other readers, Bard and I used to be coworkers at IBM.)

Got any books on point-set topology that you could recommend? I don’t know much of anything about it, and the topology books that I do have focus primarily on manifold-based stuff.

1. --bill

I learned topology from Munkres’ Red Book–I think the title was `Topology’.

1. Tinyboss

It’s teal these days.

12. Lily

This is great I love topology.

13. jofr

Just an idea, what do you think, can metaphors be described in terms of topology, for example as a kind of topological change or phase transition ?