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t w i t t e r t h r e a d : POINTLESS TOPOLOGY
Pointless topology is beloved because of its absurd name, but what on earth is it, and how do you do topology without points? A thread.
Pointless topology is beloved because of its absurd name, but what on earth is it, and how do you do topology without points? A thread.
First off, let f : X→Y be a function. Then f induces two new maps: the direct image function, also called f, which maps subsets A of X to their image f(A) under f, and the inverse image function which takes subsets B of Y to their inverse images under f.
So we have three functions,
f: X→Y,
f:P(X)→P(Y), and
f⁻¹:P(Y)→P(X).
Where P means & #39;the power set of& #39;
f: X→Y,
f:P(X)→P(Y), and
f⁻¹:P(Y)→P(X).
Where P means & #39;the power set of& #39;
Suppose that X and Y are topological spaces, and write Ω(X) and Ω(Y) for the topologies on X and Y, respectively. Then the condition that f is continuous is exactly the condition that f⁻¹ restricts to a map Ω(Y)→Ω(X) !
If f is a homeomorphism, then f is open so the direct image function restricts to a map Ω(X)→Ω(Y) - but this map is inverse to f⁻¹:Ω(Y)→Ω(X), which tells us that a homeomorphism is not just a bijection between points, but induces a bijection between the /topologies themselves/
I think that’s super cool: the structure preserving maps in this situation don’t just ‘preserve structure’ but map *directly between structures*. Pointless topology is the approach to topology where the ‘default thing’ is maps between topologies rather than maps between points.
Pointless topology approaches topologies like group theory approaches permutation groups. What are the things that make a topology a topology? First and foremost, a topology is a poset, ordered by inclusion.
∅ is a lower bound, and the whole space is an upper bound. The union of a collection (Uₐ) of open sets is the smallest set containing each of the Uₐ, i.e., the supremum (wrt ⊂) of the set {Uₐ}. Similarly, the intersection of U and V is the infimum of the set {U, V}.
So an ‘abstract topology’ ought to be a poset with an upper bound, a lower bound, a supremum for every set and an infimum for every finite set. By analogy for the set operations, we denote the supremum operation by ∨ and the infimum operation by ∧.
∨ and ∧ are called meet and joint (respectively, or antirespectively, I can never remember). Finally, in order to ensure that the elements actually act somewhat like sets, we need a distributive law.
Definition: Such posets are called locales! These are the objects of pointless topology. A topology is always a locale. A homomorphism between locales is an order preserving map that respects the meets and joins. Inverse image maps are locale homomorphisms.
Indeed, there is a natural functor Ω from topological spaces to locales given by ‘forgetting’ the set of points (X ↦ Ω(X)) and sending a continuous function to its inverse image function. Note that this is a contravariant functor.
Having established the definitions, we have to ask: how do these actually relate to topological spaces? I.e., do we have a “Cayley’s Theorem”, and if not, how close can we get? To establish this, we need to use points. No, seriously.
See, pointless topology isn’t completely free of points. Rather, the points just aren’t a ‘primary notion’; they’re not given as a part of the definition of a space, but they’re still there, as a structural aspect.
Let • denote a one point space. A point x of a topological space X corresponds to a unique map •→X, namely the map sending the point of • to x. The topology Ω(•) on • is the locale 𝟐:={∅, •}. Thus, applying Ω, points •→X correspond to locale homomorphisms Ω(X)→𝟐.
Definition: let X be a locale. A locale homomorphism X→𝟐 is called a point of X, and the set ‘spectrum’ of points of X is denoted Spec(X). If f:Y→X is a locale homomorphism, then f induces a continuous map Spec(X)→Spec(Y) by sending the point x:X→𝟐 to the point x∘f:Y→𝟐
If x is a point of a topological space, then Ω(x) is the map Ω(X)→𝟐 where Ω(U)=∅ if x∉U and Ω(U)=• if x∈U. Thus if U is an element of a locale, we say that a point X→𝟐 “inhabits” U if it sends U to •. Let I(U) be the set of points inhabiting U.
The map I is a locale homomorphism X→P(Spec(X)), from which it follows that its image is a topology on Spec(X)! Moreover, the map induced on points by a locale homomorphism is continuous
So Spec is a contravariant functor from locales to topological spaces!
One can show that the spectrum of a locale is a sober topological space. Now, (locale) points don’t correspond /exactly/ to (topological) points: the natural map X→Spec(Ω(X)) is surjective, but not in general injective. If X is sober, however, this map is a homeomorphism!
So a topological space is the spectrum of a locale if and only if it is sober. Now for the dual question: when is a locale ‘spatial’, i.e., the topology of a topological space? Recall the map I:X→P(Spec(X)). If I is injective, then it is an isomorphism of locales
If we inspect the definition, I can be seen to be injective if and only if every element U∈X is determined by which points it is inhabited by. If this holds, we say that X “has enough points”.
Thus if X has enough points, X≅Ω(Spec(X)). One can prove that if X is isomorphic to the topology of some space, then that space is Spec(X), and I is an isomorphism. That is, a locale is spatial if and only if it has enough points.
If X and Y have enough points, then f:Y→X is determined by its action on points, and every continuous map Spec(X)→Spec(Y) is the ‘point map’ of some locale homomorphism.
Compiling all these facts, we have: Spec is a contravariant equivalence between the category of locales with enough points and the category of topological spaces! This theorem is often called Stone Duality.
There are a number of arguments in favor of locales: interesting locales with no points, nice theorems about dense sublocales, etc, but this thread is already super long. Mostly I just think they’re neat, and a great way to think about topology!
This has been a thread. I can write about those things just mentioned if anyone is interested. Have a nice day
