## What Topological Spaces are π₀?

This was a fun question I thought about once. My answer is at the end, in case you’d like to try solving the problem yourself. The question is likely more interesting than my solution.

A well known theorem says that every group occurs as for some topological space . It’s not a hard construction; is a CW-complex with a 1-cell for each generator and a 2-cell for each relation. You can generalize this to higher if you want (although you only get abelian groups). Sadly, little attention is paid to lower .

is in many ways the easiest homotopy “group” to understand. Following the general definition, is the set of homotopy classes of maps from into . is 2 disconnected points . If we send to a fixed base point (which one we choose won’t matter), then there is a homotopy between iff there is a path from to ; thus is the set of path components of (this is the definition I usually use). has no group structure, and is usually just understood as a set. Obviously every set occurs as the set of path components of a topological space; namely a discreet space, so this appears trivial.

However, has a bit more structure than just a set. Path componenthood is an equivalence relation on a topological space, so the set of path components is a quotient of a topological space, and such sets come with a quotient topology (this is the finest topology so that the quotient map is continuous). So, really, is a topological space. For nice spaces (e.g. CW-complexes), is always a discreet space. However, some not-too-pathological examples, can have some interesting topology.

**Examples:
Topologist’s sin curve**: Let be the closure of the graph of , so . We want to compute .

- has 3 path components, corresponding to , , and . I will call them , respectively.
- From the definition of the quotient topology, a set is open iff the union of it’s members is open. So, for instance:
- is open, since is an open subset of
- is not open, since any open set containing a point in also contains points in

- The topology is completely described by saying and are the only open points

**Long line**: Let be the long ray with a point at infinity, that is endowed with the order topology (using lexicographic order).

- has 2 path components:
- For any two finite points and , the interval is isomorphic to for some countable ordinal . An easy transfinite induction shows is isomorphic to , and that isomorphism defines a path from to .
- On the other hand, no path connects the point at infinity to any finite point; if such a path existed, it would send to a dense subset of . In particular, such a set is cofinal (has a member bigger than any fixed element), but has uncountable cofinality (cofinality is the smallest cardinality of a cofinal set)

- The set of finite points is open in , so this corresponds to an open point in . is not open in , so it’s not open in . The resulting space is sirpinski space.

**If avoiding spoilers, read no further**

**Theorem: **Every topological space occurs as , for some other topological space .

As an informative special case, let’s build so is the trivial topology on ; that is, the only open sets are and .

First, fix a well ordering of , and let be the corresponding ordinal (or any to have big enough cofinality will do). We will give the topology where only sets of the form are open. This is equivalent to taking intervals to be basic open sets (i.e. upwards closed sets are open), as every open set must have a minimal element. The following lemma might give some intuition for this topology.

**Lemma:** With this topology, is path connected:
proof: We need to understand when a function is continuous. By definition, is continuous if is open for every . This means is open in . Notice is uniquely determined by the sequence , as is the minimum so , so a path in this space is easy to picture as a length< shrinking sequence of open sets.
To find a path from to , with , we can use the sequence of open sets where

Each path component of will be homeomorphic to a copy of with this topology; we will call the components and , and . Open sets in are defined by

- is open
- If and are non-empty open subsets of and , respectively, then is open in .

It not hard to see this defines a topology, and and are homeomorphic to in the subspace topology. This means and are both path connected, so has at most 2 path components. It remains to show:

- and are distinct path components. That is, there is no path from a point in to a point in .
- Neither nor are open in .

For 1, we will make use of the largeness of . Suppose is a path with and . Let and . Clearly ; we will get a contradiction by showing and are both open.

**Lemma:** is bounded in .
proof: Since , . However, has cofinality bigger than by König’s theorem, so any size continuum set is bounded in

Let be an upper bound for . Let . open in , and since , . As is continuous, this shows is open. A symmetric argument shows is open, thus and are both clopen, and [0,1] is disconnected.

2 is more straighforward; any open set in containing also contains points from , thus any open set in containing also contains . This completes our special case.

For the general case, fix a topological space , and we will build a space such that . Let be a disjoint copy of for each . The space will be . We define a topology on by:

- For every which is open and family of non-empty open sets, the set is open in

Notice our first example was a special case of this; the only possibilities for were and . As before, this does define a topology, and each is a path connected subspace homeomorphic to .

To see each is a distinct path component, suppose is a path with and for some . Let . We’ll show is clopen in , but clearly and , so this will be a contradiction.

As before, we know for any , is bounded in . Let . To see is closed, let . Since is open in , is an open set in , so is open. To see is open, let . is also open, and , so is open. Thus must be clopen, so such a path can’t exist.

Now we know these path components are distinct, so , so as a set is . To see the map is a homeomorphism, we need to show is an open set in iff is an open set in ; this follows immediately from the definition.

Nice post. In the first two examples you give, the spaces are Hausdorff, while your general construction only gives non-Hausdorff spaces. I think one can get any finite space occurring as of a closed subspace of using the following construction. First, consider as a directed graph with vertices given by the points of and edges given as follows: for any points , attach an edge from to if lies in the closure of . Note that this graph determines the space. Since is a finite graph, it embeds into . Replace every vertex with a small closed interval, and replace every edge from to with a compact topologist’s sine curve, attaching the boundary point to the source of the edge and the boundary line to a small subinterval of the target of the edge. The resulting closed subspace of then satisfies . (This is essentially a generalization of your first example, in which is the Sierpinski space.) By gluing arbitrary path-connected CW complexes to the small intervals, one can find such with arbitrary homotopy groups for the path components, answering Grisha’s question when is finite.

So anyway, it seems that every finite space is of a Hausdorff (even normal) space. The above construction probably also applies to countable posets with an appropriate local finiteness condition. Do you know whether in fact every space is for some Hausdorff (or normal) ?

This was a construction I tried early on in the problem, but I have yet to find a way to make get arbitrary infinite spaces as for Hausdorff . I don’t know of any reason it shouldn’t be possible; I just can’t find a way to do it.

In place of a topologist’s sin curve, there are a few other spaces you can use to connect points. You can connect points on your graph with long lines; it won’t embed in anymore, but it may be easier to picture. You can also use a space like . This example is due to my friend Sam. Of course, you’d have to embed the graph in instead to make this work.

Some ad hoc clever arguments can get you particularly nice spaces as of a Hausdorff space. For instance, we can get as follows:

Playing with similar arguments, you might be able to build any CW complex, although I haven’t found a way to get this way yet.

Another sort of ad-hoc argument I found was, if you consider the order topology on (so like the really really long line), the path components correspond to partitions of at ordinals of the form , so the resulting topology is isomorphic to the order topology on . I think a similar game can get you many order topologies, although I haven’t pursued this idea much. I should note this is likely a special case of your construction for “locally finite” topological spaces, if the assumptions you need are spelled out appropriately.

Beautiful construction for . To avoid the extra step of identifying points, one could also let .

Rather than make a new construction for , I claim that for arbitrary spaces , we have via the natural map from left to right. The fact that the map is a bijection is straightforward. To show that it is continuous, it suffices to show that each map is continuous, and using the universal property of quotient maps, this follows from the fact that is continuous. It remains to show that the map is open; for this, it suffices to note that the projection maps from a product are open, and that under the projection a path component of is sent onto a path component of .

So in particular, once one has the construction for , one obtains Hausdorff with for any cardinal . One also obtains Hausdorff with any subspace of by taking the preimage under , so for example all normal spaces occur as for Hausdorff (Urysohn). In particular, one gets all CW complexes. Note that this construction will produce regular but not necessarily normal , since infinite products of normal spaces need not be normal, and also subspaces of normal spaces need not be normal. (I forgot how delicate normality is as a condition; probably it is too much to ask that be normal, but it would be interesting to know whether one could actually *prove* that normal cannot occur for some particular space.)

In fact, I believe one can get all spaces using this idea, but there is a small gap at the end of my argument. First, note that maps from to the Sierpinski space correspond to open subsets of , so the map gives rise to an injective map , where is the largest quotient of (i.e., identify points which cannot be distinguished topologically). Since this latter map is always continuous and open onto its image, it identifies as a subspace of . If denotes with the indiscrete topology, then the map is a homeomorphism onto its image (where is the identity). Thus if and are Hausdorff spaces satisfying and then one can obtain a subspace of satisfying . So, using the long line or the topologist’s sine curve for , we have reduced, amusingly, to showing that any indiscrete space may be obtained as of a Hausdorff space. (Even without this, the above proves it for any space.)

I think I have a proof of this last statement, but there are some details I haven’t checked and this comment is already very long. If you see a nice construction, I would be interested to know it.

This is a very nice argument. I’m glad to know there’s a completely different way to solve this problem.

Getting the indiscrete topology on is not that hard, given the machinery you have from earlier. Consider the complete directed graph (so edges go both ways) with as vertices, and replace the edges with long lines, with the endpoints identified with the vertices (or an analogous construction with sin curves can work). It’s not hard to see this works.

Lovely. I had a similar idea to get indiscrete spaces but implemented it in a rather complicated way; your construction is very clean. Thanks for the cool problem!

A nice result! I wonder is there any relation between homotopy groups of different components if is not discrete? For example, do they form a sheaf over ? Or can we construct an arbitrary with arbitrary fibers?

I’m guessing there is not much communication between higher homotopy groups based in different path components, even if isn’t discrete. In particular, I think we can construct with arbitrary with such that the fibers of the path-connectedness projection have arbitrary (based, path-connected) weak homotopy type. As a lemma, we note that Brian’s space is weak contractible (I’m actually not sure about this; non-CW complexes are hard for me!). Then wedge-sum each fiber with a (based, path-connected) space of your choice!

There are some problems with taking wedge sums of <<bad>> spaces. I am not sure that the homotopy groups will behave as expected. However, I’m not sure about anything that is not CW either 🙂

I ran through the details and I’m fairly sure nothing Sam said is wrong, but I know it’s easy to wave your hands with abstract nonsense and say false things. Here are the details behind Sam’s claims.

is indeed weakly contractible. Taking as our base point, to see is trivial, we need to show that every map homotopes to the constantly 0 map. Such a homotopy should be a function with the properties:

We can let be pretty stupid:

To check that is continuous, we need to show that is open for each . We have 2 easy cases:

In fact, we didn’t use any properties of here; it turns out is strongly contractible by a very similar argument

A similar sort of trick shows you don’t have issues taking wedge sums with CW complexes. Let be a space with base point . Suppose we have representing a homotopy class. Let be defined by if and if . is the composite of with the wedge sum of the identity on with the map contracting to 0, so is also continuous. I claim that and are homotopic by the following homotopy:

We can check is continuous: let be open.

Clearly is open, as is continuous. because if then , since is the only neighborhood of in . Therefore, we don’t add any new points by saying

This is a union of open sets, hence open, so is continuous. It follows that

By the way, you have a typo in the link to Konig’s theorem

good catch; fixed now

I would like to point out that the topology on is used when is a spectral space, in which case is a profinite set. See, for example, Bhatt-Scholze, The pro-étale topology for schemes, Section 2.1.