The classical result of Sierpinski says that a compact connected Hausdorff space cannot be expressed as a countable union of non-trivial, pairwise disjoint, closed sets.

In this implies that a closed, bounded and connected subset cannot be decomposed in a countable union of nontrivial pairwise disjoint closed sets. A natural question arises: Can we construct a connected subset of Euclidean space which decomposes into countably many pairwise disjoint closed sets? I stumbled a few days ago on [this nice question]((https://math.stackexchange.com/questions/4883752/must-a-countable-disjoint-union-of-closed-balls-in-mathbbrn-with-positive/4890659#4890659).

Give this problem a try! For me it was a very nice exercise. I haven’t seen it in any textbook, which is quite suprising. Post a comment if you come up with a different construction!

This post starts with a short description of my approach to the problem which first resulted in an example in instead of a plane. We also make a historical note with an example from the 20s. After this my construction is described. I call it “Sun set”.

The first approach and historical note

First, I’ve come up with an example in . Construction is fairly simple but wordy. Here are the pictures of it (third dimension, here depth, is only to make bridges to avoid intersections we see on 2D projection of this set).

Here are the details of this construction, if you are interested in it.

I didn’t manage to embed this construction in the plane since I needed an extra dimension to create the “bridges” to avoid crossings that would destroy the pairwise disjointness. This is why I like this problem. It “fights” with me, some clever trick is needed.

I was dissatisfied with my 3D example. After failure after failure to embed it into aplane I tried out different ideas. This is how I have found the Sun Set. It has a different, more chaotic, feel than the first example. Whatever works!

I describe it in the coming section. But basically it is a union of line segments with one endpoint on the unit sphere, and the other endpoint getting closer and closer to the origin.

A few days later I found that this question existed in the early days of topology. Anna Mullikin, in the 1922 paper “Certain Theorems Relating to Plane Connected Point Sets”, constructs a Mullikin nautilus. It is formed by taking a union of infinitely many polygonal chains. The first three are pictured below (screenshot from the original paper).

I was delighted to see this neat example. She has done something I couldn’t — her example has a similar feel to my first example in . But, with a clever shape, she obtained the needed limit points to “connect” parts without the necessity of my overengineered “3D bridges”.

The virtually same set was construced by Kuratowski (see the second volume of his textbook titled “Topology” or “Continuum theory An introduction” by Nadler). I thank my friend Julia who showed me this reference.

Sun set

Here is a description of my example of a connected subset of a plane that can be written as a coutable union of non-trivial, pairwise dijoint, closed sets.

Let be a dense subset of the unit sphere with . For each define the line segment And let . Finally, let . It clearly is a union of closed, pairwise disjoint, connected sets.

Theorem. is connected.

Proof. Let be open disjoint sets with . Without loss of generality assume that contains . Since is connected, it must be contained in . It suffices to show that live in as well.

First, note that there is a natural number for which the ball is contained in the open set . Consequently, for each the endpoint of , namely , is close enough to the origin to be contained in . Hence, all with are in (since every is connected).

It remains to show that are also in the open set . Pick any index . By construction . Aiming at contradiction assume that lives in the open set . Then there is a neighborhood of with no points in common with . This is impossible because contains which is dense in the unit sphere. Hence belongs to , and so is in (’s are connected).

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