Basic Idea: the name of the game

As the name suggests, the basic idea of point-free topology is to get rid of the points. In general topology, one usually has a set, say $X$ , and wants to embed it with a structure that allows us to talk about points being close or far away. We do not really need to know the distance. The study of general topology already enables us to talk about being close or far away without the need of a distance (also called a metric). After all, when using a metric, we do talk about taking an arbitrary $\varepsilon > 0$ , but we do not really care for its value, as long as it is “big enough” (strictly greater than $0$ ). When we say $\forall \varepsilon > 0$ , we are saying:

no matter how small it may be, if it is big enough.

Denote by $\mathcal{V}(x)$ the neighbouhoods of a point $x \in X$ . A neighbourhood of $x$ is a set that is big enough to contain a ball centered at $x$ . An open set is a set that is a neighbourhood of all its points. What makes open sets really useful in this context is the fact that by knowing the open sets, it is possible to reconstruct the neighbourhoods of any point. The open sets that contain $x$ form a basis for $\mathcal{V}(x)$ . This is an important requirement (axiom, if you wish) $\mathcal{V}(x)$ must satisfy in a topological space.

Everything seems to be tied to the idea of points and the so called open sets. If we know the open sets, we can reconstruct the neighbourhoods… as long as we have a point. Having a point, we just need to take the open sets this point belongs to…

The name of the game

The game is called point-free! So, we do have the family of open sets: $\tau$ . But we do not treat them as sets. We do talk about inclusion: $A \subset B$ . But only when they are both open. We shall not say that $A$ is a subset of $B$ , or that $A$ is contained in $B$ , or that $B$ contains $A$ . Those will not be treated as sets… they do not have things that belong to them. We shall say that $A$ precedes $B$ , and denote it by

$\begin{equation} A \prec B. \end{equation}$

We wear those blinders and, from the family of open sets, all we see is a partially ordered set. A poset, for short.

Attention: I shall use greater and smaller to mean greater or equal and smaller or equal. To exclude equality I will say strictly greater or strictly smaller.

Just to organize things, instead of saying an open set, let’s agree to call it an element (of this poset). They are not sets anymore! They are elements in a poset $\tau$ .

For $x \in X$ … let’s call them points. Those are the ones we are trying to get rid of.

We do not talk about inclusion of subsets of the whole space $X$ . We do not even talk about $X$ !!! Only the open sets exist… and they are not sets! They are elements in a poset $\tau$ . 😄

The empty element, it is not “empty” because there is nothing inside it. We are not talking about sets 😎… In terms of the order in the poset, the empty element is simply the smallest element. Some like to call it $0$ . The same way, the “whole space” $X$ is not a set of points. It is simply the greatest element in the poset. We can call it $1$ .

We do talk about the union. If $\mathscr{A}$ is a family of open sets, their union is also an open set:

$\begin{equation} A = \bigcup \mathscr{A}. \end{equation}$

But since we are not talking about sets 😎… it is the supremum of $\mathscr{A}$ : the smallest element that follows (comes after) every element in $\mathscr{A}$ . We do not call it union, though. We call it the join or the supremum of elements in $\mathscr{A}$ , and write

$\begin{equation} A = \bigvee \mathscr{A}. \end{equation}$

The axiom (in general topology) that states that an arbitrary union of open sets is open translates to the fact that the supremum always exists in the poset $\tau$ .

A long time ago… when we used to deal with points 😎… it was this ability to take arbitrary unions (the supremum), that allowed us to talk about the interior of some $B \subset X$ :

The greatest among all open sets contained in $B$ .

The same way, since finite intersections of open sets are open, finite intersections of elements $A_1, \dotsc, A_n$ will correspond to the greatest element that precedes all those $A_j$ . We will call it the infimum or the meet and denote it by

$\begin{equation} A_1 \wedge \dotsb \wedge A_n. \end{equation}$

Again, not every poset is such that $a \wedge b$ is defined. However, in a topology $\tau$ , since $\tau$ is closed by finite intersections, the finite meet is defined. Taking the infimum of a finite family will be exactly the same as taking the intersections. But, since we are not actually talking about sets… 😎

And finally, although the arbitrary intersection of a family $\mathscr{A}$ of open sets might not be open, we can still talk about the infimum of those elements. Let

$\begin{equation} \mathscr{B} = \{U \in \tau :\, \forall A \in \mathscr{A},\, U \prec A\} \end{equation}$

be the elements that precede every member of $\mathscr{A}$ . The family $\mathscr{B}$ is not empty, because $\emptyset$ is the smallest element. The infimum of $\mathscr{A}$ is simply the supremum of $\mathscr{B}$ : \begin{equation} \bigwedge \mathscr{A} = \bigvee \mathscr{B}. \end{equation} In the case of point set topology — if we were talking about points… 😎 — the arbitrary meet is the interior of the intersection:

$\begin{equation} \bigwedge \mathscr{A} = \mathrm{int}\left(\bigcap \mathscr{A}\right). \end{equation}$

A poset with finite meets and finite joins is called a lattice. If we can take arbitrary meets and arbitrary joins, it is called a complete lattice.

Except for arbitrary meets, our lattice structure directly translates:

point set	point-free
$\subset$	$\prec$
$\bigcup$	$\bigvee$
$\cup$ (finite)	$\wedge$ (finite)

Be careful!

Care must be taken because despite the fact that you can always take infimum and supremum, they are not minimum and maximum. The supremum or infimum of elements that satisfy some condition might not satisfy the same condition. For example, the infimum of all real numbers that are strictly greater than $0$ is not strictly greater than $0$ .

In a $T_1$ space $X$ , given $x \in X$ , the intersection of all elements in $\mathcal{V}(x)$ is not empty… because it contains $x$ . However, looking at the open neighbourhoods of $x$ , their meet in the complete lattice of open sets is the empty set unless $x$ is isolated. In particular, it does not contain $x$ .

We can talk about the interior of a set $B \subset X$ . We take the supremum of all open sets contained in $B$ . And because the supremum is just the union, the supremum is also contained in $B$ . However, we cannot define some sort of exterior of $B$ as the least open set that contains $B$ . We can take the infimum, it is true. However, the infimum might not contain $B$ . This infimum is not the minimum.

Terminology

Well, I do not know much about the subject, yet. But I will try to stick with the following standard:

Use meet or join for a few elements: $a \wedge b$ and $a \vee b$ .
Use infimum and supremum for a family: $\bigwedge \mathscr{A}$ or $\bigvee \mathscr{A}$ .
Words like bigger and smaller mean “or equal”, just like the symbol $\prec$ .

Another example of a complete lattice

All topologies

When studying topological spaces one soon demonstrates:

Given a set $X$ , an arbitrary intersection of topological spaces over $X$ is a topological space.

This allow you to have, infimum and supremum in the family of all topological spaces over $X$ . The smallest element is the caotic topology: $\{\emptyset, X\}$ . The greatest element is the discrete topology: $\mathcal{P}(X)$ . The infimum is just the intersection: the open sets that are common to all those topologies. And the supremum can be realized as the infimum among all those that are bigger. Since there is a greatest element, this family of those that are bigger is non-empty.

If you take the infimum of all topologies that contain a family $\mathscr{C}$ of subsets of $X$ , since the infimum is just intersection, it does contain $\mathscr{C}$ . It is the smallest topology over $X$ that contain $\mathscr{C}$ .

Be careful, again! In $\mathbb{R}$ , if $\tau$ is the infimum of all topologies where $\frac{1}{n}$ converges to $0$ , is it true that $\tau$ is the smallest topology such that $\frac{1}{n} \rightarrow 0$ ? It seems not!.

All groups, and more…

In a group $G$ , take the lattice of all subgroups. When studying groups, you learn very soon that arbitrary intersection of subgroups is a subgroup. Just the same way, you have a complete lattice. In particular, you have the subgroup generated by a set of elements of $G$ .

Do the same for vector spaces, measurable spaces, rings, fields, algebras, etc…

The challenge

Instead of saying we are forbidden to talk about points, we talk about a complete lattice that might be or not isomorphic to the topology of a topological space $X$ .

Given a set $X$ and a complete lattice $L$ :

Can we construct a topology $\tau$ over $X$ such that $\tau$ is (lattice-)isomorphic to $L$ ?
Is this $\tau$ unique?
What if we do not have $X$ … only $L$ ? Can we find an $X$ ? Will it be unique?

Let $\tau$ be a topology in $X$ and $\gamma$ a topology in $Y$ . Given a $\tilde{f}: \gamma \rightarrow \tau$ :

Can we construct a continuous $f: X \rightarrow Y$ such that $\tilde{f} = f^{-1}|_{\gamma}$ ?
Is it unique?

A related question…

Given a lattice $L$ , what are the families that could possibly correspond to $\mathcal{V}(x)$ for a certain point $x$ in a some topological space whose topology is isomorphic to $L$ ?

In “Be careful!”, we saw that this is not as easy if you have the lattice $L$ but not the sets. For example, the family $\{(-1/n, 1/n)\}$ generates $\mathcal{V}(0)$ in $\mathbb{R}$ , but the family $\{(0, 1/n)\}$ generates a (non-principal!) filter strictly larger than $\mathcal{V}(0)$ . How do you distinguish them only looking at the lattice?

Those questions are answered in the first chapter of this book:

📚 Jorge Picado & Aleš Pultr Frames and Locales: topology without points. Springer Basel, Series Frontiers in Mathematics, Vol. 28, xx + 400 pp., 2012 (ISBN: 978-3-0348-0153-9).

Continues… next post.