It sounds like jargon from a bad episode of Star Trek. But it’s a real thing. It’s a monstrous object that lives in the plane, but is impossible to draw.
Do you want to see how snake-like it is? Okay, but beware… this video clip is a warning:
This snake-like monster is also called the ‘pseudo-arc’. It’s the limit of a sequence of curves that get more and more wiggly. Here are the 5th and 6th curves in the sequence:
Here are the 8th and 10th:
But what happens if you try to draw the pseudo-arc itself, the limit of all these curves? It turns out to be infinitely wiggly—so wiggly that any picture of it is useless.
In fact Wayne Lewis and Piotr Minic wrote a paper about this, called Drawing the pseudo-arc. That’s where I got these pictures. The paper also shows stage 200, and it’s a big fat ugly black blob!
But the pseudo-arc is beautiful if you see through the pictures to the concepts, because it’s a universal snake-like continuum. Let me explain. This takes some math.
The nicest metric spaces are compact metric spaces, and each of these can be written as the union of connected components… so there’s a long history of interest in compact connected metric spaces. Except for the empty set, which probably doesn’t deserve to be called connected, these spaces are called continua.
Like all point-set topology, the study of continua is considered a bit old-fashioned, because people have been working on it for so long, and it’s hard to get good new results. But on the bright side, what this means is that many great mathematicians have contributed to it, and there are lots of nice theorems. You can learn about it here:
• W. T. Ingraham, A brief historical view of continuum theory,
Topology and its Applications 153 (2006), 1530–1539.
• Sam B. Nadler, Jr, Continuum Theory: An Introduction, Marcel Dekker, New York, 1992.
Now, if we’re doing topology, we should really talk not about metric spaces but about metrizable spaces: that is, topological spaces where the topology comes from some metric, which is not necessarily unique. This nuance is a way of clarifying that we don’t really care about the metric, just the topology.
So, we define a continuum to be a nonempty compact connected metrizable space. When I think of this I think of a curve, or a ball, or a sphere. Or maybe something bigger like the Hilbert cube: the countably infinite product of closed intervals. Or maybe something full of holes, like the Sierpinski carpet:
or the Menger sponge:
Or maybe something weird like a solenoid:
Very roughly, a continuum is ‘snake-like’ if it’s long and skinny and doesn’t loop around. But the precise definition is a bit harder:
We say that an open cover 𝒰 of a space X refines an open cover 𝒱 if each element of 𝒰 is contained in an element of 𝒱. We call a continuum X snake-like if each open cover of X can be refined by an open cover U1, …, Un such that for any i, j the intersection of Ui and Uj is nonempty iff i and j are right next to each other.
Such a cover is called a chain, so a snake-like continuum is also called chainable. But ‘snake-like’ is so much cooler: we should take advantage of any opportunity to bring snakes into mathematics!
The simplest snake-like continuum is the closed unit interval [0,1]. It’s hard to think of others. But here’s what Mioduszewski proved in 1962: the pseudo-arc is a universal snake-like continuum. That is: it’s a snake-like continuum, and it has continuous map onto every snake-like continuum!
This is a way of saying that the pseudo-arc is the most complicated snake-like continuum possible. A bit more precisely: it bends back on itself as much as possible while still going somewhere! You can see this from the pictures above, or from the construction on Wikipedia:
• Wikipedia, Pseudo-arc.
I like the idea that there’s a subset of the plane with this simple ‘universal’ property, which however is so complicated that it’s impossible to draw.
Here’s the paper where these pictures came from:
• Wayne Lewis and Piotr Minic, Drawing the pseudo-arc, Houston J. Math. 36 (2010), 905–934.
The pseudo-arc has other amazing properties. For example, it’s ‘indecomposable’. A nonempty connected closed subset of a continuum is a continuum in its own right, called a subcontinuum, and we say a continuum is indecomposable if it is not the union of two proper subcontinua.
It takes a while to get used to this idea, since all the examples of continua that I’ve listed so far are decomposable except for the pseudo-arc and the solenoid!
Of course a single point is an indecomposable continuum, but that example is so boring that people sometimes exclude it. The first interesting example was discovered by Brouwer in 1910. It’s the intersection of an infinite sequence of sets like this:
It’s called the Brouwer–Janiszewski–Knaster continuum or buckethandle. Like the solenoid, it shows up as an attractor in some chaotic dynamical systems.
It’s easy to imagine how if you write the buckethandle as the union of two closed proper subsets, at least one will be disconnected. And note: you don’t even need these subsets to be disjoint! So, it’s an indecomposable continuum.
But once you get used to indecomposable continua, you’re ready for the next level of weirdness. An even more dramatic thing is a hereditarily indecomposable continuum: one for which each subcontinuum is also indecomposable.
Apart from a single point, the pseudo-arc is the unique hereditarily indecomposable snake-like continuum! I believe this was first proved here:
• R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51.
Finally, here’s one more amazing fact about the pseudo-arc. To explain it, I need a bunch more nice math:
Every continuum arises as a closed subset of the Hilbert cube. There’s an obvious way to define the distance between two closed subsets of a compact metric space, called the Hausdorff distance—if you don’t know about this already, it’s fun to reinvent it yourself. The set of all closed subsets of a compact metric space thus forms a metric space in its own right—and by the way, the Blaschke selection theorem says this metric space is again compact!
Anyway, this stuff means that there’s a metric space whose points are all subcontinua of the Hilbert cube, and we don’t miss out on any continua by looking at these. So we can call this the space of all continua.
Now for the amazing fact: pseudo-arcs are dense in the space of all continua!
I don’t know who proved this. It’s mentioned here:
• Trevor L. Irwin and Salawomir Solecki, Projective Fraïssé limits and the pseudo-arc.
but they refer to this paper as a good source for such facts:
• Wayne Lews, The pseudo-arc, Bol. Soc. Mat. Mexicana (3) 5 (1999), 25–77.
Abstract. The pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. It is, however, also the most important, being homogeneous, having several characterizations, and having a variety of useful mapping properties. The pseudo-arc has appeared in many areas of continuum theory, as well as in several topics in geometric topology, and is beginning to make its appearance in dynamical systems. In this monograph, we give a survey of basic results and examples involving the pseudo-arc. A more complete treatment will be given in a book dedicated to this topic, currently under preparation by this author. We omit formal proofs from this presentation, but do try to give indications of some basic arguments and construction techniques. Our presentation covers the following major topics: 1. Construction 2. Homogeneity 3. Characterizations 4. Mapping properties 5. Hyperspaces 6. Homeomorphism groups 7. Continuous decompositions 8. Dynamics.
It may seem surprising that one can write a whole book about the pseudo-arc… but if you like continua, it’s a fundamental structure just like spheres and cubes!