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.

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

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

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.

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

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.

Sheaves in Geometry and Logicand Peter Johnstone’sTopos Theory(the old 1971 book, not theElephant) 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 theStone Spacesbook, on the whole. $\endgroup$ – Peter LeFanu Lumsdaine Feb 1 '11 at 22:04