Martin EscardoApr 27, 2024

What is a topological space?

It is a mathematical device to define what a continuous function is, in a general setting.

1. A topological space is a set X together with a collection of subsets of X, called open, such that finite intersections of open sets are open, and arbitrary unions of open sets are open.

2. A function of topological spaces is continuous if inverse images of open sets are open.

What is the intuition behind (1) and (2)?

I claim that it is better to ask, instead, how mathematicians came up with (1) and (2).

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Show threadMartin Escardo

At the beginning, people considered continuous functions ℝ → ℝ.

One of the wrong intuitions, at that time, was that such a function is continuous if "you can draw it without lifting the pen".

A counter-example is Cantor's devil's staircase. This continuous function you can't draw, with or without lifting the pen.

https://en.wikipedia.org/wiki/Cantor_function

Mathematicians spoke of continuous functions for a long time before there was a precise definition of continuous function. This was a vague idea, which, nevertheless, was useful.

At some point the definition of continuity for a function f : ℝ → ℝ was elucidated.

∀ x ∈ ℝ , ∀ ε > o, ∃ δ > 0 , ∀ x' ∈ ℝ, ∣ x - x' | < δ → | f(x) - f(x') | < ε.

This definition allowed a lot of theorems to be proved rigorously. This is why it was useful.
A lot of theorems that were claimed, could now be proved.
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Cantor function - Wikipedia

Apr 27, 2024 at 6:13pmWeb
Show threadMartin EscardoApr 27, 2024

This definition *does* have an intuition.

Suppose you want to calculate f(x₀), for example because you are an engineer and want to build a bridge, where x₀ is a physical quantity. Unfortunately, we can't measure x₀ exactly.

But you still want to know what y₀ = f (x₀) is, at least approximately.

But "approximately" doesn't mean anything. You want to know y₀, say, with two correct decimal digits. This will do to build a robust bridge.

Then the question is, how many correct digits of x₀ do you need to know, in order to get the desired two decimal digits of y₀?

More generally, we want the following to be the case. In order to know n digits of output of the function, it is enough to know m digits of the input, where m depends on the input and on n.

This is possible if and only if the function is continuous.

So one intuition about continuity is that "finite amounts of output depend on only finite amounts of input".

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Show threadMartin EscardoApr 27, 2024

But this was for functions ℝ → ℝ only.

Soon people started to consider continuous functions "of two variables", that is, functions ℝ × ℝ → ℝ, and then of n variable, so ℝⁿ → ℝ, and in each case a different definition was needed.

Moreover, because people were trying to e.g. solve differential equations, which amounts to given one *input* function (called the initial condition) to figure out an *output* function (the solution), people came across functions mapping continuous functions to continuous functions, which, themselves, may or may not be continuous.

So a general definition of continuity was needed to clean up the mess and be able to make progress more efficiently.

A first, rather useful, general setting was that of a metric space. You say what the distance between two things (e.g. real numbers, tuples of real numbers, continuous functions) is. The axioms for metric spaces are very intuitive, and I won't reproduce their statement here.

But I want to say this.

1. The definition of continuity is the same as above, with the absolute value of the difference replaced by the distance function d. A function f : X → Y is continuous iff

∀ x ∈ X , ∀ ε > o, ∃ δ > 0 , ∀ x' ∈ X, d(x,x') < δ → d(f(x),f(x')) < ε.

2. We can then define a set U to be open if for every x ∈ U there is ε > 0 such that every x' with d(x,x') < ε is in U.

3. The open sets are closed under finite intersections and arbitrary unions.

4. A function is continuous in the above ε-δ sense iff inverse images of open sets are open.

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Show threadMartin EscardoApr 27, 2024

This, and much more, is beautifully explained in the book

George F. Simmons. Topology and modern analysis
https://archive.org/details/introduction-to-topology-and-modern-analysis-simmons

which I highly recommend.

What happens next is that metric spaces are not enough. There are sets on which we want to consider continuous functions which can't be metrized so that metric continuity coincides with the notion of continuity we want.

I like this book because it starts from trivial things, making them less and less trivial as we progress, until it gets eventually to many things, including Stone duality. I self-learned topology from this book as an undergrad.

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Introduction to topology and modern analysis : Simmons, George Finlay, 1925- : Free Download, Borrow, and Streaming : Internet Archive

This is a classical introductory textbook on topology and analysis

Internet Archive
Show threadMartin EscardoApr 27, 2024

Then comes computer science. Smyth, Abramsky and Vickers propose a radically different intuition for the axioms of topology and the definition of continuous function.

I will stop this post at this point, at least for now, giving four seminal references.

Michael .B. Smyth. Power domains and predicate transformers: a topological view.
https://link.springer.com/chapter/10.1007/BFb0036946

Michael .B. Smyth. Topology.
https://dl.acm.org/doi/10.5555/162573.162536

Samson Abramsky. Domain Theory and the logic of observable properties
https://arxiv.org/abs/1112.0347

Steve Vickers. Topology via logic.
https://www.cambridge.org/gb/universitypress/subjects/computer-science/programming-languages-and-applied-logic/topology-logic?format=PB&isbn=9780521576512

I started my PhD after reading the above, and I was lucky to have had Smyth as my supervisor.

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Power domains and predicate transformers: A topological view

The broad theme of the paper is that topological concepts are basic to computer science. Such concepts as “specifications”, “predicate transformer”, and “nondeterminism” can be greatly illuminated by being formulated in topological...

SpringerLink
Show threadMartin EscardoApr 27, 2024

But one should say the following. Actually, the connection of topology with computation started with Brouwer, who was a topologist himself, in fact one of the founding fathers of topology before Hausdorff come up with the axiomatization of topological spaces which I described above.

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Show threadMartin EscardoApr 27, 2024

And finally, in the same way metric spaces were not enough, and so we moved from metric spaces to topological spaces, Grothendieck found that topological spaces were not enough for him, and so he needed, more generally, toposes.

And recently condensed sets and pyknotic sets have also been proposed as a convenient generalization of the notion of (topological) space.

It is easy to lose track of the original intuitions. It is also easy for new intuitions to replace the original ones, when we move to more general settings. And we always move to more general settings in mathematics.

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Show threadMartin EscardoApr 28, 2024

Going back to the original notion of topological space, it is worth mentioning that it can be axiomatized in several equivalent ways.

1. The above, with open sets.
2. With neighbourhood systems.
3. With closure operators.
4. With interior operators.
5. With Moore-Smith convergence of nets (where nets generalize sequences; sequences are indexed by the natural numbers, whereas nets are indexed, more generally, by directed sets).
6. More things omitted here.

One can say that each of the above emphasize different intuitions.

However, I think they actually emphasize different proof techniques.

In practice, when one does classical topology, one freely switches between the above definitions, according to which one seems to give the best proof, or at least the proof the author is more comfortable with.

In any case, I think it is "wrong" to say that the notion of topological space (given by any of the above equivalent axiomatizations) comes from a single intuition.

There are several intuitions, including intuitions that existed before the above notion of topological space was formulated, and intuitions that emerge when we work with topological spaces.

To repeat, in order to understand topology (and its generalizations) one needs to understand the history.

Or at least the story.

G.F. Simmons tells a very good story in the book I cited in (5/)

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Show threadMartin EscardoApr 28, 2024

And I'll also briefly discuss the computational intuition for the open-set axiomatization, proposed by Smyth, Abramsky and Vickers.

They say, respectively, that open sets correspond to

(1) Semi-decidable properties.

(2) Observable properties.

(3) Affirmable properties.

Let me tell the story based on (2), and you can try to figure out the story based on (1) if you know computability theory. Regarding (3), the closed sets are the refutable properties, and if you know about Popperian philosophy, you may try to reconstruct the intuition yourself.

Suppose a machine is outputting, forever, the decimal expansion of some number, digit by digit, and you are observing its output.

For example, at some point you see "3.1415" and are still waiting for the next digit.

If you could open the machine and examine its program, perhaps you would be able to prove or disprove that it is programmed to output the decimal expansion of π.

But the machine is given as a blackbox to you, and you only see its output.

What can you observe about its output? For example in this case you can observe that it starts with "3.1415" and therefore the machine is definitely not computing the decimal expansion of 𝑒.

The observable properties are closed under finite conjunctions. To observe that the conjunction of finitely many properties holds, you just observe each of them, one at a time.

The observable properties are closed under arbitrary disjunctions. To observe that a disjunction of arbitrarily many properties holds, you need to observe only one of them.

Of course, this is just intuition. I look at this in more detail in my monograph "Synthetic topology of data types and classical spaces".

https://www.sciencedirect.com/science/article/pii/S1571066104051357

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Show threadMartin EscardoApr 28, 2024

This last intuition proves, once again, that there isn't a single source of intuitions for the notion of topological space.

You need to learn the intuitions as you learn topology and its applications. It is unlikely that anybody can see the intuitions by staring at the axioms, whatever axiomatization is taken as primary.

(From the point of view constructive mathematics, the axiomatization based on open sets seems to be primary. But this would be a discussion for a separate thread.)

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Show threadflApr 28, 2024
@MartinEscardo @MartinEscardo @MartinEscardo I don't "stare" at the axioms. And I know that topological spaces can be defined in several ways. But one must also take into consideration the psychology of one's students when one teaches them a concept. And defining something in the most cryptic and esoteric way (open sets) is certainly not the best way to adress their need to connect the ideas together and make sense of them. Using R X R plus neighborghoods plus Bolzano-Weierstrass theorem just as Weierstrass did historically is better than using open sets. And it is also the occasion to show how mathematicians make their definitions: they choose a biimplication among several ones for opportunity reasons (here the shortness of the open sets definition I think).
Show threadTTimoApr 29, 2024
@MartinEscardo great thread, thanks!
Show threadJoshua GrochowApr 28, 2024
@MartinEscardo do you know, was Grothendieck's issue essentially that the Zariski topology just doesn't have very many open sets? Or was there something more to it?
Show threadBoardersApr 28, 2024
@joshuagrochow @MartinEscardo I believe Serre noticed something like the fact that G bundles are not classified by cohomology classes, but they are if you pull back by étale maps instead
Show thread⏚ Antoine Chambert-LoirApr 28, 2024

@boarders @joshuagrochow @MartinEscardo
Indeed, it was already a surprise when Serre proved that the Zariski topology could recover the sheaf cohomology of coherent sheaves ,(FAC and GAGA papers), but that did not furnish the “correct” cohomology groups that would help proving the Weil conjectures (along the way suggested by Weil himself).

At some point, Serre observed that “algebraic fiber spaces” were not locally trivial for the Zariski topology, but we're made trivial after passing to an étale covering.
He made a talk at the Chevalley seminar about it

Grothendieck was there and, at the end of the talk, told Serre that they had (« on a ») the Weil cohomology.

http://www.numdam.org/item/SCC_1958__3__A1_0/

Espaces fibrés algébriques

Show threadJohn Carlos BaezApr 28, 2024

@antoinechambertloir @boarders @joshuagrochow @MartinEscardo - there is an amusing interplay between the words "cover" and "covering" in mathematics. As I started learning topology I thought an "open cover" was something completely different from a "covering space" - indeed, strangely, it never occurred to me to wonder if these concepts were related. But then, learning concepts like "etale map" and "etale topology", and learning that an open cover \(U_\alpha\) of a space \(X\) gives a local homeomorphism \(\sum_\alpha U_\alpha \to X\), I realized that open covers and covering spaces are deeply related. Of course this is something that all experts already know.... but I became curious why people don't run around saying "hey! coverings and covers both give etale maps!"

I have no idea what Serre and Grothendieck and others thought about "covers versus coverings" - some of this may be special to English-language math - but Serre's observation that "algebraic fiber spaces" can be trivialized by pulling back along an étale covering sounds like exactly the right reason to develop algebraic geometry in a way that emphasizes their relationship.

Show thread⏚ Antoine Chambert-LoirApr 28, 2024

@johncarlosbaez @boarders @joshuagrochow @MartinEscardo
There is no such phonetic proximity in french, revêtement et recouvrement are quite different words. However, both Serre and Grothendieck had thought a lot about topology, having thought about fiber spaces in particular.

In his PhD thesis, Serre generalized the concept of Hurewicz fibration to what we now call Serre fibration— requiring the lifting property for maps from cubes only.

The first work of Grothendieck in algebraic geometry was in 1955, about fiber spaces.

https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf

Show thread⏚ Antoine Chambert-LoirApr 28, 2024

@johncarlosbaez said “I have no idea whether Serre and Grothendieck and others thought of the etale topology as a kind of generalized topology that unified the concept of covering space and open cover.”

I should read again Serre's 1958 talk on fiber spaces, but I'm certain the mood was there. They wanted that Weil cohomology theory!

Clearly they did and stated it as such already in SGA 4 (1964, théorie des topos...). Descent theory was another part of the plan, already there around 1960.

@boarders @joshuagrochow @MartinEscardo

Show thread⏚ Antoine Chambert-LoirApr 28, 2024

@johncarlosbaez
Actually I never remember what is a cover and what is a covering!!!

@boarders @joshuagrochow @MartinEscardo

Show threadJohn Carlos BaezApr 28, 2024

@antoinechambertloir - It's even more confusing because in English mathematicians say "covering space" (never "cover space") and "open cover" (never "open covering") where the word is being used as an adjective, but "SU(2) double covers SO(3)", where the word is being used as a verb.

@boarders @joshuagrochow @MartinEscardo

Show threadflApr 27, 2024
@MartinEscardo In a book called "Grundzüge der Mengenlehre" ("Elements of Set Theory") I love that.
Show threadflApr 27, 2024
@MartinEscardo Hausdorff defined topologies in terms of neighborhoods (not open sets) and it was Weierstrass who invented the concept.
Show threadflApr 27, 2024
@MartinEscardo In between Cantor gave his opinion.
Show threadflApr 27, 2024
@MartinEscardo The problem (Weierstrass's) at first was more limit points than continuity. Continuity came later.
Show threadflApr 27, 2024
@MartinEscardo It's a problem by the way because giving the definition of topology as open sets is the best way to generate in student's mind the question : "why did they use this definition?" The answer is easy: topology was not defined as open sets at the beginning. Neighborhoods are more appropriate for an intuitive definition.
Show threadflApr 27, 2024
@MartinEscardo And (horresco referens) it is Bourbaki who propelled the definition of topologies as open sets. Take out a clove of garlic.
Show threadflApr 27, 2024
@MartinEscardo To conclude using neighboorhoods and giving the Bolzano-Weierstrass theorem is certainly the best way to keep topology in its historical order. The problem then is to find an example easy enough to explain what the Bolzano-Weierstrass theorem can be good for.
Show threadTobyBartelsApr 28, 2024
@fl @MartinEscardo : He also required them to be Hausdorff (hence that name), yes?
Show threadflApr 28, 2024
@TobyBartels @MartinEscardo I don't remember but according to Wikipedia, yes. It was he who defined them in the same book.
Show threadTobyBartelsApr 28, 2024

@fl @MartinEscardo : Not just that he defined Hausdorff spaces in the same book where he defined general topological spaces, but that his most general definition in that book was of only Hausdorff spaces.

At least that is what Wikipedia seems to say, and that is what the definition does that I found on page 213 of a 1914 edition of his book on the Internet Archive. (But I didn't look through the rest of the chapter to see if he has a more general definition later on.)

Show threadflApr 28, 2024
@TobyBartels @MartinEscardo OK.
Show threadTobyBartelsApr 28, 2024
@MartinEscardo : But open-set spaces aren't enough either; for example, convergence in measure can't be defined by open sets. (Of course, this is convergence, not continuity, but they're interdefinable, and you could tell an equivalent history based on convergence instead of continuity. In any case, I'm sure that somebody has considered maps between function spaces that preserve convergence in measure, in other words continuous maps between such spaces, but I'm no measure theorist, so I don't know what they are.)
Show threadTobyBartelsApr 28, 2024
@MartinEscardo : (This is also a different sense of not enough, one closer to the origins of the topic in analysis, than the sense that leads to locales and toposes etc.)
Show threadrntzApr 27, 2024

@MartinEscardo

> In order to know n digits of output of the function, it is enough to know m digits of the input, where m depends on the input and on n.

if we demand that this dependency be computable (that from a desired output precision n I can compute the necessary input precision m), does this change the meaning of continuity?

Show threadazulApr 27, 2024

@rntz @MartinEscardo Yes, it changes the notion of continuity. There are classically continuous functions that don't have any computable modulus of continuity (a function that tells you how much delta you need for a given epsilon).

(EDIT: This example is bad, see replies.) E.g. constant function that returns Chaitin's constant, because Chaitin's constant does not have a computable fast Cauchy sequence. (It has a computable Cauchy sequence that does not converge at a computable rate.)

Show threadazulApr 27, 2024
@rntz @MartinEscardo This isn't a very good example though ... it kinda doesn't have anything to do with continuity. Maybe a better example would be a continuous function that uses digits of the input to construct Chaitin's constant gradually (via dovetailing).
Show threadazulApr 27, 2024
@rntz Ok I think I've got an example: consider f : [0,1] -> [0,1] where f(x) is defined by looking at the decimal expansion of x (preferring .xxx50000... over .xxx4999...). f(x) is defined as the sum C_i for all i where the ith digit of x is a 5. We define C_i as the contribution to Chaitin's constant obtained during the ith step dovetailing all TMs. So usually C_i is 0, but sometimes it's a nonzero value. This f is continuous because C_i gets arbitrarily small (noncomputably) as i increases.
Show threadazulApr 27, 2024
@rntz argh, no I think this still doesn't work because f is not continuous around .xxx5000... / .xxx4999...) if the corresponding C_i is nonzero. I wonder if there's a way to fix this🤔
Show threadazulApr 27, 2024
@rntz me designing a step function: "as you can see this is clearly continuous"
Show threadazulApr 28, 2024

@rntz ok, i was trying to be cute by using the halting probability to come up with an example, but it's very hard to do anything with TMs without introducing discontinuity.

but now I think i've come up with an example that actually works:

let f(x) be the definite integral from 0 to 1/x of the busy beaver function as a step function on the reals.
then f is continuous, but its modulus of continuity around f(1/n) gives us an upper bound on BB(n-1), so it's not computable.

Show threadrntzApr 27, 2024
@typeswitch @MartinEscardo interesting! Do you have any other examples? I think technically the Chaitin's constant constant function does not qualify as an example, because I didn't insist on the function itself being computable, only the "modulus" (I guess? not familiar with how to use that word correctly); so for any constant function (even non-computable), we require 0 bits of input to determine (but not compute) the entire output. Am I mistaken?
Show threadazulApr 27, 2024
@rntz Yes, I think my first example was just bad actually. I tried to simplify the construction I had in mind (see the other reply) and then later realized i had simplified it too much.
Show threadMartin EscardoApr 27, 2024
Side remark. The devil's staircase is computable. I haven't thought whether we can define it constructively if we take the reals to be the Dedekind reals, and we don't have any choice axioms. But my hunch is that we can.
Show threadChristina GrossackApr 27, 2024

@MartinEscardo

The wikipedia definition you linked looks like a perfectly good constructive definition of a function from [ℕ,3] to [ℕ,2] (where I'm writing [A,B] for functions A to B, since unicode doesn't have an ℕ or an ω in the exponent). Indeed, you can easily write it in haskell using lists instead of functions from ℕ.

Then the usual map sending [ℕ,2] to the unit interval should be fine, so the only question is whether we can write a base-3 expansion of every real in the unit interval...

The short answer is "no", I think, but the longer answer (due to Vickers, iirc?) is "yes". In case the map [ℕ,3] to the unit interval is a proper surjection (which it should be) then we can (constructively!) define a map from the unit interval by defining it on [ℕ,3] instead (as long as we check the map we define respects the "equivalence relation" on [ℕ,3]).

This is basically because proper surjections of locales are coequalizers of their kernel pair. So even though not every element of the unit interval has a base-3 expansion, we can ~reason~ about the unit interval as though it were true!

I think this should be enough to get a constructive definition of the cantor function, and maybe I'll try to write it down precisely later tonight

Show threadsoaprootApr 27, 2024
@MartinEscardo Draw it with a pen in the sense of "topological curve" as defined at https://en.m.wikipedia.org/wiki/Curve or in some other sense?
Curve - Wikipedia