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As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a limited understanding. As such, I have some questions:

  1. What are some accessible texts or online references on the subject?
  2. What are some recent results in point-free topology that are unique to the subject, i.e., not translations of results from general topology into localic language?
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You might be interested in the relevant part of maths.gla.ac.uk/~tl/cafe_topos_intro.pdf . –  Qiaochu Yuan Feb 1 '11 at 0:09
I recommend reading Johnstone's quick survey "The point of pointless topology" to get a feel for the difference between locales and topological spaces and for why it might be worthwhile to deal with locales. It's meant as a "trailer" for his book "Stone spaces", which Andrej recommends in his answer. –  Omar Antolín-Camarena Feb 1 '11 at 2:19
"Topology via Logic" by Vickers is quite elementary and does everything in terms of locales. It is written for computer scientists with no previous knowledge of topology, so it is probably the most accessible introduction to the subject available. –  Michael Greinecker Feb 1 '11 at 7:31
This doesn’t quite fit what you ask for, but much of what I know about locales, I learnt from reading books about Topos Theory. Mac Lane and Moerdijk’s Sheaves in Geometry and Logic and Peter Johnstone’s Topos Theory (the old 1971 book, not the Elephant) both include some very good bits of exposition on locale theory, though unfortunately (if you’re mainly interested in just the locales, not the toposes) they’re a bit buried among all the topos theory. I actually found those both more helpful than the Stone Spaces book, on the whole. –  Peter LeFanu Lumsdaine Feb 1 '11 at 22:04
@Qiaochu: thanks for the plug. As I've now moved university, a safer link is arxiv.org/abs/1012.5647. –  Tom Leinster Jan 9 '13 at 5:27
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3 Answers

I would recommend Peter Johnstone's "Stone Spaces", Cambridge University Press, 1982.

For a recent result see Alex Simpson's "Measure, Randomness and Sublocales". He shows that in locale theory it is possible to have an isometry-invariant measure on $\mathbb{R}^n$ for which all subsets are measurable. He also defines the locale of random sequences as the sublocale of those sequences which satisfy all measure 1 properties. The locale of random sequences is not empty (but has no points!), and in fact its measure is 1. All of this is quite impossible if you insist that spaces must have lots of points.

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@Michael: the paper is not using "measure" in the usual sense. Also, the correct term is "piqued." –  Qiaochu Yuan Feb 1 '11 at 0:06
What Alex does is compatible with choice, i.e., he does in in ZF I think. No need to play any set-theoretic or logical games, just plain cool math –  Andrej Bauer Feb 1 '11 at 0:08
@Andrej: Thankyou for the Simpson link, it looks lovely! One quick question on your summary of it: when you say “In locale theory it is possible to have a measure…”, do you mean “…one can define a measure…”, or “…it is consistent that there exists a measure…”? –  Peter LeFanu Lumsdaine Feb 1 '11 at 5:04
Perhaps it's worth pointing out what has changed in this approach. Because we add new parts to spaces, it may turn out that what used to be a cover is not a cover anymore. In fact, this is what happens. For example, the Banach-Tarski "paradox" does not go through anymore because the paradoxical "decompostion" of the unit ball is not a decomposition anymore. –  Andrej Bauer Feb 1 '11 at 12:04
@Peter: So you're now calling me a set-theorists? We'll settle this in Oberwolfach. –  Andrej Bauer Feb 2 '11 at 8:29
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This isn't a precise answer to either of your two questions. However, it sounds like you're interested in learning about locales, so maybe it's useful to make the following general point.

The theory of locales is often motivated as follows: often in topology (e.g. in the definition of sheaf) the points of a space are irrelevant, so we might as well abstract them away and work with open sets only. That's fine, but what possibly doesn't get said often enough is that the resulting theory is a piece of algebra.

Let me say that more exactly. A frame is a partially ordered set with finite meets and arbitrary joins, such that meets distribute over joins. Equivalently, it is a set X equipped with:

  • a binary operation $\wedge: X^2 \to X$ and a constant $\top \in X$ (thought of as the top or greatest element)
  • for each set I, an I-ary operation $\bigvee_I: X^I \to X$

satisfying a bunch of equations. (There's no need to mention the order relation explicitly, since it can be recovered from the rest of the structure: $x \leq y$ iff $x \wedge y = x$.) A map of frames is a map of sets commuting with all the operations. Thus, the category of frames is a category of algebras in any of several standard senses: e.g. it's monadic over the category of sets.

(It's a slightly unusual category of algebras in that it includes infinitary operations, and indeed infinitary operations of arbitrarily high arity, but still, it shares many of the good features of old friends like the categories of groups, rings, modules, etc.)

The category of locales is by definition the opposite of the category of frames.

So, this is a really literal instance of the slogan "geometry is dual to algebra".

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Interesting, so in some sense, the notion of frame (for lack of a better word) flattens out the algebra of open sets? And the corresponding notion of locale is meant to model the structural properties of the topological spaces which would give rise to a specific frame? –  Michael Blackmon Feb 2 '11 at 17:59
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My favorite reference for point-free topology is the very new book.

Frames and Locales: Topology Without Points by Picado and Pultr.

This book is an excellent book for those who want to learn about point-free topology for the first time and as a reference for those who are already familiar with point-free topology.

As for recent results in point-free topology, I have recently been researching a duality in point-free topology. My new duality represents all zero-dimensional frames as Boolean algebras along with specified least upper bounds.

We therefore define a Boolean admissibility system to be a pair $(B,\mathcal{A})$ such that $\mathcal{A}$ is a subset of the powerset $P(B)$ that satisfies the following properties.

  1. If $R\in\mathcal{A}$, then $R$ has a least upper bound.

  2. $\mathcal{A}$ contains each finite subset of $P(B)$

  3. If $R\in\mathcal{A},S\subseteq B,S\subseteq\downarrow\bigvee R=\{a\in B|a\leq\bigvee R\}$ and $R$ refines $S$(i.e. for each $r\in R$ there is an $s\in S$ with $r\leq s$), then $S\in\mathcal{A}$ as well.

  4. If $R\in\mathcal{A}$ and $R_{r}\in\mathcal{A},\bigvee R_{r}=r$ for $r\in R$, then $\bigcup_{r\in R}R_{r}\in\mathcal{A}$

  5. If $R\in\mathcal{A}$, then $\{r\wedge a|r\in R\}\in\mathcal{A}$ for each $a\in B$.

Property $1$ states that $\mathcal{A}$ is a collection of least upper bounds and properties $2-5$ state that $\mathcal{A}$ contains all sets with least upper bounds that you would want to include. For instance, in a Boolean algebra you would always want to include the least upper bound of a finite set. Axioms $2-5$ get rid of all the trivial differences between Boolean admissibility systems. A Boolean admissibility system $(B,\mathcal{A})$ is called subcomplete if whenever $R\cup S\in\mathcal{A}$ and $r\wedge s=0$ whenever $r\in R,s\in S$, then $\bigvee R$ exists.

I recently proved that the category of Boolean admissibility systems is equivalent to the category of all pairs $(L,A)$ such that $L$ is a frame and $A$ is a Boolean sublattice of $L$ which is a "basis" for $L$(i.e. $A$ is a sublattice of $L$ consisting of complemented elements where each element in $L$ is the join of elements in $A$). This equivalence of categories restricts to an equivalence between the category of all zero-dimensional frames and subcomplete Boolean admissibility systems.

With this duality, I was able to characterize point-free topological properties in terms of the corresponding Boolean admissibility systems. These properties include ultraparacompactness, ultranormality, $\kappa$-compact zero-dimensional frames(where $\kappa$ is a cardinal), extremally disconnected frames(as Boolean admissibility systems which are complete Boolean algebras), Lindelof $P$-frames(as $\sigma$-complete Boolean algebras), and other properties.

This result does not have as much of a pointed analogue since very rarely does a Boolean admissibility system correspond to zero-dimensional space (i.e. a spatial zero dimensional frame). The Boolean admissibility systems that correspond to topologies are precisely the subcomplete Boolean admissibility systems $(B,\mathcal{A})$ where each ideal closed under taking least upper bounds in $\mathcal{A}$ can be extended to a maximal ideal closed under taking least upper bounds in $\mathcal{A}$. This property can be characterized by a very strong distributivity property and very few Boolean admissibility systems satisfy this property.

I should also note that one can represent any pair $(L,A)$ where $L$ is a frame and $A$ is a "basis" for $L$ as the poset $A$ along with specified least upper bounds. Unfortunately, even though this setting is more general, I have not yet found a way to represent any separation axioms in terms of posets with specified least upper bounds.

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