Interpretations of Topological Space as a Heyting Algebra

Question

I have recently learned about Heyting algebras which I find quite fascinating, as I am more intuitionistically inclined. One of the main examples of Heyting algebras are given by topological spaces as follows:

Let $(X,\tau)$ be a topological space. For $U,V\in\tau$, define $$U\wedge V:=U\cap V,\hspace{.5cm} U\vee V:=U\cup V, \hspace{.5cm}U\Rightarrow V:=\mathrm{Int}(U^c\cup V),\hspace{.5cm}1:=X,\hspace{,3cm}0:=\varnothing$$ And of course, as usual we define $\neg U:=U\Rightarrow 0$.

Then it turns out that $(\tau,\wedge,\vee,\Rightarrow)$ forms a Heyting algebra! This fact alone is quite interesting, but I was wondering if we can go further. Most references I could find online simply use topological spaces as an example and stop investigations after showing $\tau$ is a Heyting algebra.

My question is this:

Can we make a dictionary that translates between properties of topological spaces and properties/statements about logic?

On my own, I came up with a few that are quite easy to see:

$\begin{align*} \neg U&=X-\overline{U}&(\neg U=0)&\Longleftrightarrow U\text{ is dense}\\\neg\neg U&=\mathrm{Int}(\overline{U})&(\neg\neg U=0)&\Longleftrightarrow U\text{ is nowhere dense}\\U\vee\neg U&=X-\partial U&(U\vee \neg U=1)&\Longleftrightarrow U\text{ is clopen}\\&&(U\Rightarrow V=1)&\Longleftrightarrow U\subseteq V \end{align*}$

What about other topological properties that we know and love? What does it mean about the corresponding Heyting algebra if $X$ is compact or Hausdorff or Regular or path connected, etc? What about continuous maps between topological spaces and all of the properties they might have? What do those imply about the induced morphisms between the Heyting algebras? Could we, for example, transport the definition of the fundamental group through this correspondence to get something meaningful in terms of the Heyting algebra?

Any thoughts or references would be greatly appreciated!

Answer 1

When one views the topology of a topological space as a lattice the most natural thing to do is focus on it being a complete lattice with finite meets, arbitrary joins, and where meets distribute over arbitrary meets. Any such lattice is automatically a Heyting algebra. But, the Heyting algebra structure is stronger. To be precise, one should always consider morphisms. The morphisms of topological spaces are the continuous functions. When viewing topologies as lattices a continuous function $f\colon X\to Y$ induces a function $g\colon \tau_Y \to \tau_X$ in the opposite direction given by the inverse image function of $f$. This function $g$ preserves the meets and joins but not the Heyting algebra structure. So, one introduces distinct terminology. The lattice of a topological space is most naturally viewed as a frame, namely a complete lattice in which meets distributive over arbitrary joins. A frame homomorphism is then required to preserve just finite meets and arbitrary joins. This gives rise to the category $\mathbf{Frm}$ of frames. The observations above simply state that there is a functor $\mathbf{Top}\to \mathbf {Frm}^\mathrm{op}$, and the latter category is $\mathbf{Loc}$, the category of locales. The category $\mathbf {Hey}$ of Heyting algebras has the same objects as $\mathbf {Frm}$ and as $\mathbf {Loc}$ but different morphisms; those that also preserve the implication relation.

This comment should indicate a refinement of your question. As you may anticipate your question does indeed have very well-established answers. Further to the excellent suggestions in the comments I'll add Johnstone's "Stone Spaces".