Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

February 20, 2019

Tychonoffication

Posted by John Baez

Joshua Meyers is a grad student in my real analysis class. We had an interesting conversation about topology and came up with some conjectures. Maybe someone has already proved them. I just want to write them down somewhere.

First some background. For every topological space XX there’s a set C(X,[0,1])C(X,[0,1]) consisting of all continuous functions from XX to [0,1][0,1]. And there’s a natural map from XX into [0,1] C(X,[0,1])[0,1]^{C(X,[0,1])}. The ‘cube’ [0,1] C(X,[0,1])[0,1]^{C(X,[0,1])} is the product of copies of [0,1][0,1], one for each continuous function from XX to [0,1][0,1]. So, it gets the product topology. The natural map sends each point XX in XX to the function sending each continuous function f:X[0,1]f: X \to [0,1] to f(x)f(x).

Get it? Maybe an equation will help. The natural map

i:X[0,1] C(X,[0,1])i: X \to [0,1]^{C(X,[0,1])}

is defined by

i(x)(f)=f(x)i(x)(f) = f(x)

This is a standard ‘role reversal’ trick, turning a function into the argument and the argument into the function.

By Tychonoff’s theorem, the cube [0,1] C(X,[0,1])[0,1]^{C(X,[0,1])} is a compact Hausdorff space. In my real analysis class I had the kids show that if XX is compact Hausdorff, the map is an embedding of XX into the cube [0,1] C(X,[0,1])[0,1]^{C(X,[0,1])}. That is, the image of ii with its subspace topology is homeomorphic to XX.

So, every compact Hausdorff space is homeomorphic to a subspace of a ‘cube’: a product of copies of [0,1].

That’s the background. Then Joshua Meyers told me that more generally, a Tychonoff space can be defined as a space that’s homeomorphic to a subspace of a cube. A Tychonoff space needs to be Hausdorff (since a cube is), but it doesn’t need to be compact (since you can embed an open interval in a cube). The usual definition of Tychonoff space looks complicated and arbitrary, but it’s equivalent to this one; the definition I gave should be the definition, and the usual definition should be a theorem.

Now for the interesting part. Any space has a ‘Tychonoffication’. That’s not a very pretty word, but it’s extremely easy to guess what it means!

Namely, given any space X, we define the image of

i:X[0,1] C(X,[0,1])i: X \to [0,1]^{C(X,[0,1])}

with its subspace topology, to be the Tychonoffication of XX. It’s obviously a Tychonoff space, and there’s obviously a continuous map from XX to its Tychonoffication, namely i:Xim(i)i: X \to im(i).

This leads to:

Conjecture. Let TychTych be the category of Tychonoff spaces and continuous maps. The forgetful functor

U:TychTop U: Tych \to Top

has a left adjoint

F:TopTych F: Top \to Tych

and this left adjoint is Tychonoffication. The above map from any topological space to its Tychonoffication is the unit of the adjunction. TychTych is a reflective subcategory of TopTop.

We went further and guessed that this adjunction factors as the composite of two other adjunctions. For this, note that any continuous map between topological space f:XYf: X\to Y factors as

Xfim(f)jY X \stackrel{f}{\longrightarrow} im(f) \stackrel{j}{\longrightarrow} Y

But we can give im(f) two different topologies. One is the quotient topology coming from the surjection f:Xim(f)f: X \to im(f). Another is the subspace topology coming from the inclusion j:im(f)Yj: im(f) \to Y.

These topologies don’t need to be the same: after all, the quotient topology knows nothing about the topology of YY, while the subspace topology knows nothing about the topology of XX. Indeed, suppose XX is the real line \mathbb{R} with its discrete topology, YY is \mathbb{R} with its codiscrete topology, and ff is the identity function. Then im(f)=im(f) = \mathbb{R}. With the quotient topology it’s \mathbb{R} with its discrete topology, while with the subspace topology it’s \mathbb{R} with its codiscrete topology!

There is always a continuous map from im(f)im(f) with its quotient topology to im(f)im(f) with its subspace topology. We can apply this when ff is the natural map

i:X[0,1] C(X,[0,1]) i: X \to [0,1]^{C(X,[0,1])}

im(i)im(i) with its subspace topology is the Tychonoffication of XX. But we can also give im(i)im(i) its quotient topology. Different points x,yXx, y \in X will get mapped to the same point of im(i)im(i) iff these points are not separated by any continuous function f:X[0,1]f: X \to [0,1], i.e. we cannot find ff with f(x)f(y)f(x)\ne f(y).

Now, a completely Hausdorff space is a topological space where any two distinct points xx and yy are separated by a continuous function f:[0,1]f: \to [0,1]. So, it seems that im(i)im(i) with its quotient topology is the ‘complete Hausdorffication’ of X.

So, Joshua Meyers and I seem to believe something like this. There are three categories: Top,TychTop, Tych, and CompHausCompHaus, the category of completely Hausdorff space and continuous maps.

Conjecture. There are functors

TychU 1CompHausU 2Top Tych \stackrel{U_1}{\longrightarrow} CompHaus \stackrel{U_2}{\longrightarrow} Top

with left adjoints

TopF 2CompHausF 1Tych Top \stackrel{F_2}{\longrightarrow} CompHaus \stackrel{F_1}{\longrightarrow} Tych

Given any topological space XX, F 2(X)F_2(X) is im(i)im(i) with its quotient topology, while F 1(F 2(X))F_1(F_2(X)) is im(i)im(i) with its subspace topology.

So, we’re breaking up Tychonoffication into two steps. The first step is complete Hausdorffication: it identifies points that can’t be separated by any continuous function f:[0,1]f: \to [0,1]. (By the way, these are the same as the points that can’t be separated by any continous function f:Xf : X \to \mathbb{R}.) The second step may coarsen the topology—I don’t have a really clear mental image of what’s going on here, but a suitable example should clarify it!

I am feeling too lazy to prove these conjectures, since this has nothing to do with my main line of work. So, if anyone wants to prove them—or find a proof in the existing literature—I’d be very happy! Please let me know.

Posted at February 20, 2019 5:17 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3090

26 Comments & 0 Trackbacks

Re: Tychonoffication

There’s another way to factor this adjunction (conjecturally)! We can give XX the weakest topology that makes ii continuous; let’s use XX' to denote XX with this topology. Then XX' is completely regular. You can define this condition by saying that every closed set and point can be separated by a continuous real-valued function (i.e. for each closed AA and point pAp\notin A there is a continuous real-valued f:Xf:X\rightarrow \mathbb{R} such that f| Af|_A is constantly cf(p)c\neq f(p)).

So ii then factors to XXi(X)X\rightarrow X'\rightarrow i(X). We can conjecture that the adjunction TychTop\mathrm{Tych}\leftrightharpoons \mathrm{Top} then factors as TychCompRegTop\mathrm{Tych}\leftrightharpoons \mathrm{CompReg}\leftrightharpoons \mathrm{Top} in addition to TychCompHausTop\mathrm{Tych}\leftrightharpoons \mathrm{CompHaus}\leftrightharpoons \mathrm{Top}

giving a commutative square (I would draw it but I don’t feel like trying to make \leftrightharpoons vertical).

Posted by: meygerjos on February 20, 2019 7:57 PM | Permalink | Reply to this

Re: Tychonoffication

Something looks odd to me. Did you mean to write U 1:TychCompHausU_1: Tych \to CompHaus? If UU’s are reserved for “underlying/forgetful” functors, then I would have thought U 1:CompHausTychU_1: CompHaus \to Tych, i.e., every compact Hausdorff is in fact Tychonoff.

Posted by: Todd Trimble on February 20, 2019 10:37 PM | Permalink | Reply to this

Re: Tychonoffication

CompHaus is the category of completely Hausdorff spaces, not compact Hausdorff spaces.

Posted by: meygerjos on February 20, 2019 11:19 PM | Permalink | Reply to this

Re: Tychonoffication

Okay, my bad for reading too quickly.

Posted by: Todd Trimble on February 20, 2019 11:41 PM | Permalink | Reply to this

Re: Tychonoffication

Sorry, I didn’t notice how much my category of completely Hausdorff spaces, CompHausCompHaus, looks like the category of compact Hausdorff spaces. I’m not really doing anything with compactness in this post, though it’s motivated by the theorem saying that every compact Hausdorff space embeds in a cube.

Posted by: John Baez on February 21, 2019 4:39 PM | Permalink | Reply to this

Re: Tychonoffication

The F:TopCompHausF: Top \to CompHaus is usually called the Stone-Cech compactification, and it is the closure of the image of XI hom(X,I)X \to I^{\hom(X, I)} with the subspace topology. See e.g. the nLab.

Posted by: Todd Trimble on February 20, 2019 10:50 PM | Permalink | Reply to this

Re: Tychonoffication

Just so passersby can follow what’s going on: here you’re using CompHausCompHaus to mean the category of compact Hausdorff spaces. Given a topological space XX, its image under i:X[0,1] C(X,[0,1])i: X \to [0,1]^{C(X,[0,1])} is the universal way of making it into a Tychonoff space. So you’re saying we can go further and make in into a compact Hausdorff space in a universal way by taking the closure of the image. All this is very nice!

Usually I’ve seen Stone-Cech compactification done to Hausdorff spaces, but XX doesn’t need to be Hausdorff for all this stuff to work. When we take the image of XX under ii, we’re forcing it to be Hausdorff—as part of making it Tychonoff.

Posted by: John Baez on February 21, 2019 4:48 PM | Permalink | Reply to this

Re: Tychonoffication

In any case, if YY is Tychonoff and f:XYf: X \to Y, then we have a naturality square

X i X I hom(X,I) f I hom(f,I) Y i Y I hom(Y,I)\array{ X & \stackrel{i_X}{\to} & I^{\hom(X, I)} \\ \mathllap{f}\; \downarrow & \, & \downarrow \mathrlap{I^{\hom(f, I)}} \\ Y & \stackrel{i_Y}{\to} & I^{\hom(Y, I)} }

and the image of i Xi_X as a subspace of I hom(X,I)I^{\hom(X, I)} can be identified with the equalizer of its cokernel pair. (The cokernel pair is formed by taking the pushout

X i X I hom(X,I) i X I hom(X,I) K X\array{ X & \stackrel{i_X}{\to} & I^{\hom(X, I)} \\ \downarrow \mathrlap{i_X} & \; & \downarrow \\ I^{\hom(X, I)} & \to & K_X }

in TopTop, and regarding the unlabeled arrows in this pushout as a parallel pair of maps I hom(X,I)K XI^{\hom(X, I)} \rightrightarrows K_X.) One can easily check there is a map K f:K XK YK_f: K_X \to K_Y induced by ff, and there is a serially commutative diagram

I hom(X,I) K X I hom(f,I) K f I hom(Y,I) K Y\array{ I^{\hom(X, I)} & \rightrightarrows & K_X \\ \downarrow \mathrlap{I^{\hom(f, I)}} & \; & \downarrow \mathrlap{K_f} \\ I^{\hom(Y, I)} & \rightrightarrows & K_Y }

so that by taking equalizers of the cokernel pairs, we get a commutative diagram

im(i X) I hom(X,I) K X I hom(f,I) K f Y i Y I hom(Y,I) K Y\array{ im(i_X) & \to & I^{\hom(X, I)} & \rightrightarrows & K_X \\ \downarrow & \; & \downarrow \mathrlap{I^{\hom(f, I)}} & \; & \downarrow \mathrlap{K_f} \\ Y & \stackrel{i_Y}{\to} & I^{\hom(Y, I)} & \rightrightarrows & K_Y }

where the vertical map on the left gives a factoring of ff through the surjective map q X:Xim(i X)q_X: X \to im(i_X). The uniqueness of this factoring follows from surjectivity. This proves that im(i X)im(i_X) is indeed the Tychonoffication of XX.

Posted by: Todd Trimble on February 20, 2019 11:36 PM | Permalink | Reply to this

Re: Tychonoffication

I mean, that im(i X)im(i_X) is the left adjoint of the forgetful functor TychTopTych \to Top.

Posted by: Todd Trimble on February 20, 2019 11:40 PM | Permalink | Reply to this

Re: Tychonoffication

Nice, Todd!

So it sounds like you understand something I did better than I do. Namely, in this post I noticed that any continuous map

f:XY f: X \to Y

has a natural factorization

XABY X \to A \to B \to Y

where AA is im(f)im(f) with the topology it gets from being a quotient space of XX, and BB is im(f)im(f) with the topology it gets from being a subspace of YY.

To get AA from XX we identify points but otherwise change the topology as little as possible, while to get BB from AA we don’t change the underlying set at all, but we make the topology coarser.

I was going to ask what this is an example of. (As Kennedy proclaimed, after he became a category theorist: “Ask not for an example of this — ask what this is an example of!”)

It sounds like you know, and it sounds like the answer involves the phrase “equalizer of the cokernel pair”. What’s the general story? What sort of category has this sort of 3-step factorization for every morphism?

Posted by: John Baez on February 21, 2019 10:10 PM | Permalink | Reply to this

Re: Tychonoffication

So I think that every finitely complete, finitely cocomplete category has this sort of 3-step factorization, but often you don’t see it because often that middle step collapses to an isomorphism.

Every morphism in such a category factors as a regular epi followed by a something in the middle followed by a regular monomorphism. (Side remark: in TopTop, regular monos are the same as subspace inclusions, and regular epis are the same as quotient projections.) That is, every morphism f:XYf: X \to Y factors as

Xpcoim(f)im(f)iYX \stackrel{p}{\to} coim(f) \to im(f) \stackrel{i}{\to} Y

where I define im(f)im(f) to be the equalizer of the cokernel pair of ff, and coim(f)coim(f) to be the coequalizer of its kernel pair.

To deduce the thing in the middle, let’s give the name qq to the map Xim(f)X \to im(f) which exists by the universal property of im(f)im(f) as equalizer of cokernel pair plus the fact that ff equalizes its cokernel pair. Let’s call π 1,π 2:EX\pi_1, \pi_2: E \rightrightarrows X the kernel pair of ff. I claim that qπ 1=qπ 2q\pi_1 = q\pi_2. That’s easy because we have

iqπ 1=fπ 1=fπ 2=iqπ 2i q\pi_1 = f\pi_1 = f\pi_2 = i q\pi_2

and now use the fact that ii is monic to deduce qπ 1=qπ 2q\pi_1 = q\pi_2. So then q:Xim(f)q: X \to im(f) must factor through the coequalizer of the kernel pair (π 1,π 2)(\pi_1, \pi_2), which is p:Xcoim(f)p: X \to coim(f); that factoring gives the thing in the middle coim(f)im(f)coim(f) \to im(f).

Anyway, applying these general observations to the unit i X:XI hom(X,I)i_X: X \to I^{\hom(X, I)} of the monad you introduced, the coimage of i Xi_X is this quotient topology on XX that you’re proposing as the complete Hausdorffication (and I think you’re right about that), and the image i Xi_X is of course the Tychonoffication. But I am also supposing that there is a much more general nonsense story that might be told, abstracting away from the particular topological context we are discussing. I don’t know what the story is, exactly, yet.

By the way, I’m not sure which definition of Tychonoff you meant that looks arbitrary and scary, but the nLab defines a Tychonoff space to be any space arising as a subspace of a compact Hausdorff space, and I think that’s actually a very nice flexible definition. Shot through the story here is the fact that II is a cogenerator in the category of compact Hausdorff spaces, something closely connected with the Urysohn lemma and such.

Posted by: Todd Trimble on February 21, 2019 11:53 PM | Permalink | Reply to this

Re: Tychonoffication

The “arbitrary and scary” definition of Tychonoff space is the one used on Wikipedia: a Tychonoff space is a completely regular Hausdorff space. Here a completely regular topological space is one where points can be separated by closed sets by continuous real-valued functions.

This is not actually very scary, but “a subspace of a product of closed intervals” is better and “a subspace of a compact Hausdorff space” is even better.

It’s actually great to have both the extrinsic definition (characterizing it as a subspace of something) and the intrinsic one.

Posted by: John Baez on February 22, 2019 6:05 AM | Permalink | Reply to this

Re: Tychonoffication

My favorite definition of Tychonoff space is that for any open set U𝒪(X)U\in \mathcal{O}(X) we have U={V𝒪(X)V<<<U}U = \bigcup \{ V\in \mathcal{O}(X) \mid V \lt\!\!\!\!\lt\!\!\!\!\lt U \}, where V<<<UV\lt\!\!\!\!\lt\!\!\!\!\lt U means that the closure of VV is contained in UU and that there exists W𝒪(X)W\in \mathcal{O}(X) such that V<<<W<<<UV\lt\!\!\!\!\lt\!\!\!\!\lt W \lt\!\!\!\!\lt\!\!\!\!\lt U (coinductively).

Posted by: Mike Shulman on February 22, 2019 7:39 AM | Permalink | Reply to this

Re: Tychonoffication

Admittedly, that coalgebraic perspective is pretty sweet. I never thought of approaching the Urysohn lemma from that point of view.

Posted by: Todd Trimble on February 23, 2019 2:53 AM | Permalink | Reply to this

Re: Tychonoffication

It’s a ternary factorization system. In fact, this very example is the first one on that page.

Posted by: Mike Shulman on February 22, 2019 12:20 AM | Permalink | Reply to this

Re: Tychonoffication

That’s nice! You probably remember us writing about the weak ternary factorization system in CatCat in Lectures on n-categories and cohomology. Given a functor p:EBp: E \to B we factored it thus:

We start with EE; then we throw in new 2-morphisms (equations between morphisms) that we get from BB; then we throw in new 1-morphisms (morphisms), and finally new 0-morphisms (objects). It’s like a horse transforming into a person from the head down. First it’s a horse, then it’s a centaur, then it’s a faun-like thing that’s horse from the legs down, and finally it’s a person.

But I’d never thought about the strict ternary factorization system in TopTop!

Posted by: John Baez on February 22, 2019 6:12 AM | Permalink | Reply to this

Re: Tychonoffication

This sounds like something that ought to be in Johnstone’s book Stone Spaces, but I don’t have it with me at the moment to look.

It also sounds like we ought to be saying something about codensity monads.

Posted by: Mike Shulman on February 21, 2019 6:03 PM | Permalink | Reply to this

Re: Tychonoffication

Apologies in advance for pedantry verging on sealioning, but shouldn’t it be “Tychonoffification”, just as it should be “Hausdorffification”?

(Corrections/confirmation from e.g. Blake Stacey welcome.)

Posted by: Yemon Choi on February 21, 2019 8:19 PM | Permalink | Reply to this

Re: Tychonoffication

That was exactly my first thought on seeing this post.

(And thanks, Yemon, for introducing me to the term sealioning. I remember once reading the comic that inspired the term, but hadn’t encountered the word.)

Posted by: Mark Meckes on February 21, 2019 8:56 PM | Permalink | Reply to this

Re: Tychonoffication

I think it should really be “Tychonoffifification”, just to make it easier to say.

Posted by: John Baez on February 21, 2019 9:59 PM | Permalink | Reply to this

Re: Tychonoffication

By the way, Tychonoff spaces are sometimes called T 312T_ {3\frac{1}{2}} spaces — or even T πT_\pi spaces! That’s cute, though admittedly not very rational.

Posted by: John Baez on February 22, 2019 6:47 AM | Permalink | Reply to this

Re: Tychonoffication

I see what you did there.

More seriously, despite the inherent arbitrariness of the scale, when I was first learning about the various separation axioms, I found the T nT_n terminology easier to keep straight than “Hausdorff”, “regular”, “normal”, etc. How are you supposed to remember whether “regular” or “normal” is meant to be nicer, let alone how “Hausdorff” fits in there?

Posted by: Mark Meckes on February 22, 2019 11:50 AM | Permalink | Reply to this

Re: Tychonoffication

I’ve never really managed to remember the T nT_n terminology, though I probably knew it for a few months in my youth. It’s just too unevocative for me.

I know a bunch of grad students who fall in love with point set topology and separation axioms and have to be “talked down”, like: “yes, this is fun stuff, but it’s not fashionable, since it’s hard to do anything new in this area that seriously affects other subjects…”

I don’t recall going through that phase myself.

Posted by: John Baez on February 22, 2019 5:20 PM | Permalink | Reply to this

Re: Tychonoffication

I don’t claim to be able to remember the T nT_n terminology now. But “regular” and “normal” aren’t evocative enough to tell me which is which either. It is at least easier to remember that T 4T_4 is stronger than T 3T_3 than it is to remember that “normal” is stronger than “regular”.

Fortunately, mostly I’ve never needed to remember any of this stuff — at least not since I took a qualifying exam on which I had to prove that a topological space is Tychonoff if and only if it’s homeomorphic to a subspace of a cube.

Posted by: Mark Meckes on February 22, 2019 6:58 PM | Permalink | Reply to this

Re: Tychonoffication

I didn’t get sucked into this stuff myself as a graduate students; I think it was only with nLab work did I ever take a closer look.

For me, the main virtue of the T nT_n terminology is that it serves to remind me that they all include the T 0T_0 condition, that points are distinguished by the systems of neighborhoods containing them. A uniform structure plus T 0T_0 gets you all the way to T 312T_{3\frac1{2}} (if I remember correctly), and pretty soon after that point I begin to lose interest, although the category of T 4T_4 spaces (which are normal T 2T_2 spaces) has IIRC some pretty decent categorical properties.

I think I learned a lot of this stuff indirectly through Toby Bartels. He wrote a lot of the nLab material on separation axioms.

From a career perspective John is probably right to talk down those students, but quite a lot of point-set topology is just so weird and fascinating and charming; it’s easy to understand why those grad students can get sucked in.

Posted by: Todd Trimble on February 23, 2019 2:50 AM | Permalink | Reply to this

Re: Tychonoffication

This is a very interesting post for quantum gravity theorists. We are in the business of deriving the emergent (classical) manifolds from quantum information, using categorical operads like the n-cubes operad in each dimension. Quantum logic is infinite dimensional because the cardinality of a set is replaced by the dimension of a vector space, as a rough idea. Amazing that Hilbert somehow had this deep intuition so long ago, but then much about quantum mechanics was swept under the rug in the twentieth century!

Posted by: Marni Dee Sheppeard on February 22, 2019 10:54 PM | Permalink | Reply to this

Post a New Comment