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Stream: learning: questions

Topic: motivation for topological spaces

David Egolf (Jan 18 2022 at 17:36):

Mike Shulman said:

Undergraduate topology courses (and often even graduate ones) sometimes give the impression that the open-set-based definition of "topological space" is "the correct notion of space", but in fact it's just one of a wide array of possibilities, and not a particularly well-behaved one at that.

(This thread is over my head, but this particular paragraph is interesting to me).
I've only self-studied topology, and the definition of a topology always left me a bit puzzled.
It is interesting to hear that the open-set-based definition is not necessarily the "one right way"!

Graham Manuell (Jan 19 2022 at 12:43):

David Egolf said:

I've only self-studied topology, and the definition of a topology always left me a bit puzzled.
It is interesting to hear that the open-set-based definition is not necessarily the "one right way"!

Whether open-set-based approaches are 'correct' obviously depends on what you are trying to do, but I definitely think they are the most important ones by a wide margin. You were probably right to be puzzled, since it is unlikely the sources you were reading made any attempt to motivate the axioms, but a perfectly convincing motivation does exist (in terms of verifiable properties). See here or the book Topology via Logic by Steve Vickers. I do not know any similarly convincing motivations for any other approach.

Mike Shulman (Jan 19 2022 at 15:46):

Graham Manuell said:

a perfectly convincing motivation does exist (in terms of verifiable properties). See here or the book Topology via Logic by Steve Vickers. I do not know any similarly convincing motivations for any other approach.

I would argue that that motivation actually motivates locales, not topological spaces.

Graham Manuell (Jan 19 2022 at 17:38):

I think it works as a motivation for both, though being a pointfree topologist I'm obviously going to agree that locales are better.

John Baez (Jan 19 2022 at 18:45):

I've only self-studied topology, and the definition of a topology always left me a bit puzzled.

The idea is that given a point $x$, the open sets $U$ containing $x$ consists of points that are close to $x$ in some sense. Precisely which sense - well, that's specified by $U$. If $y \in U$ we can say $y$ is "$U$-close to $x$". People don't actually say this, they just say $y \in U$, but this is what it means, and as evidence for this notice that people call $U$ a neighborhood of $x$.

John Baez (Jan 19 2022 at 18:46):

You presumably started by learning metric spaces, where you have a distance function, and you can say $d(x,y) < \epsilon$, which means "$y$ is of distance $\lt \epsilon$ from $x$".

John Baez (Jan 19 2022 at 18:47):

It turns out that metric spaces are not sufficiently general to handle all important notions of closeness, so we need to get more abstract and just talk about sets $U$ of points that are close in some sense to $x$.

John Baez (Jan 19 2022 at 18:47):

Any decent book on topology should say something like this, but I'm not sure they all do.

David Egolf (Jan 19 2022 at 19:53):

That makes sense @John Baez . The motivation from metric spaces (and their open balls) is helpful.
I seem to recall open sets being associated with "nearness" in some introduction I read, but I found that counterintuitive. It seemed wrong to me to say that any two numbers from $\mathbb{R}$ are "near" (contained in the same open set) - even numbers very far apart! However, the idea that each open set provides a different notion of nearness provides a clearer picture of what's going on.

Jens Hemelaer (Jan 19 2022 at 19:55):

I think the original definition of topological spaces by Hausdorff is a bit more intuitive (see "Grundzüge der Mengenlehre", p213). A topological space according to Hausdorff was a set of points, together with for each point a collection of neighborhoods of this point, satisfying four axioms. The four axioms were:
(A) each point has a neighborhood;
(B) if $U$ and $V$ are neighborhoods of a point, then the intersection $U \cap V$ contains a neighborhood of the point;
(C) if $U$ is a neighborhood of $x$ and $y \in U$, then there is a neighborhood $V$ of $y$ with $V \subseteq U$;
(D) if two points are distinct, then we can find disjoint neighborhoods for them.

You can then define open sets as those subsets $U$ such that for each $x \in U$ there is a neighborhood $V$ of $x$ with $V \subseteq U$. From this it follows that neighborhoods are open (so his definition of neighborhood is different from the current one), and that unions and finite intersections of open sets are again open. You don't need Axiom D (Hausdorffness) for this, so this axiom was later dropped.

Mike Shulman (Jan 19 2022 at 22:12):

I would say that the "observations" definition motivates locales, and then you get a derived motivation for (sober) topological spaces as the spatial locales.

Mike Shulman (Jan 19 2022 at 22:13):

This is certainly a great motivation. But it's not the only notion of "space" — why should the notion of "space" a priori have anything to do with observation?

Mike Shulman (Jan 19 2022 at 22:15):

Other notions of space have perfectly good motivations too. A subsequential space, or pseudotopological space, is an abstraction of the notion of convergence, which is an intuitive thing once you've been through calc 2. Functional analysts often define a "topology" by just giving its notion of convergence.

Fawzi Hreiki (Jan 19 2022 at 22:29):

Ultimately, there are two ways of defining/giving spaces: by their elements (e.g. simplicial complexes, subsequential spaces, etc...) or by their functions (locales, affine schemes, etc...). But this is more like a spectrum than a clear divide.

Fawzi Hreiki (Jan 19 2022 at 22:30):

Topological spaces are somewhat mixed: their points are defined covariantly but their open regions are defined contravariantly.

Mateo Carmona (Jan 20 2022 at 03:35):

Maybe someone finds interesting this thesis: Grothendieck et les topos : rupture et continuité dans les modes d'analyse du concept d'espace topologique. A very good summary.

Jorge Soto-Andrade (Jan 20 2022 at 04:39):

Mateo Carmona said:

Maybe someone finds interesting this thesis: Grothendieck et les topos : rupture et continuité dans les modes d'analyse du concept d'espace topologique. A very good summary.

Pas mal, merci!

Graham Manuell (Jan 20 2022 at 12:02):

Mike Shulman said:

I would say that the "observations" definition motivates locales, and then you get a derived motivation for (sober) topological spaces as the spatial locales.

Not to belabour what I think to not be a terribly important point, but I think you get more than motivation for just sober spaces. If you think of points as the things which actually exist and the opens as the observable properties, then it is not ridiculous to imagine that every property should be determined by its points (spatiality), but that certain points might be completely indistinguishable by what we can observe (giving rise to spaces which even violate the $T_0$ axiom).

Graham Manuell (Jan 20 2022 at 12:14):

Mike Shulman said:

This is certainly a great motivation. But it's not the only notion of "space" — why should the notion of "space" a priori have anything to do with observation?

This is a fair point and perhaps word 'space' is just too vague for the concept I am thinking of. Some people think of groupoids as a kind of space, though to me this is entirely distinct from the more topological notions of space that I am thinking of.

I do not think it is obvious, but I have come to believe it is true, that observable properties are important for every variant of this kind of space I am thinking of. In particular, I think that the reals should always be viewed as a locale and never as a bare set.

Now sometimes one might care about slightly different notions of space, for example metric spaces. Metric spaces are very good and it makes sense to consider a metric even a discrete set. However, I believe it is better to consider locales / topological spaces equipped with a metric structure if you want to think of the reals as a metric space. Simon Henry distinguishes between 'metric sets' for the former and '(pre)metric locales' for the latter.

As for convergence spaces, I can't help but feel these would also be best understood as a convergence structure in addition to a more topological notion. However, I concede that I do not know exactly how to get this to work. I do find the theory of convergence vector spaces to be very pretty and I hope we will one day have a pointfree version of them.

John Baez (Jan 20 2022 at 18:01):

The motivation from metric spaces (and their open balls) is helpful. I seem to recall open sets being associated with "nearness" in some introduction I read, but I found that counterintuitive. It seemed wrong to me to say that any two numbers from $\mathbb{R}$ are "near" (contained in the same open set) - even numbers very far apart! However, the idea that each open set provides a different notion of nearness provides a clearer picture of what's going on.

Right! Maybe the author was bad at explaining things. Each pair of real numbers is near compared to those that are farther away. To a cosmologist, Mars is very near the Earth.

John Baez (Jan 20 2022 at 18:03):

So "near" is not a yes-or-no business. Instead open sets provide highly flexible "standards of nearness".

John Baez (Jan 20 2022 at 18:04):

Anyway, once you get over this road-bump the best thing is to see what people do with open sets, e.g. define continuous functions and prove results about them, so you can see how they actually work.

David Egolf (Jan 20 2022 at 18:09):

John Baez said:

Anyway, once you get over this road-bump the best thing is to see what people do with open sets, e.g. define continuous functions and prove results about them, so you can see how they actually work.

In this case, I found that I was able to proceed while puzzled, and was able to work through most of an introductory book on topology. So I've done a bunch of exercises at least. That did help develop the intuition, thankfully!