TITLE: Duality between compactness and Hausdorffness
QUESTION [53 upvotes]: Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also Hausdorff. A minimal Hausdorff topology is such that no strictly coarser topology is Hausdorff.
The trivial topology is compact. Every topology that is coarser than a compact topology is also compact. A maximal compact topology is such that no strictly finer topology is compact.
A compact Hausdorff topology is both minimal Hausdorff and maximal compact.
A minimal Hausdorff topology need not be compact and a maximal compact topology need not be Hausdorff (Steen & Seebach, Examples 99 and 100).
Question:
It seems that there is some duality between Hausdorffness and compactness? Can this kind of duality be stated more explicitly (e.g. in some category theoretic formulation)? Is a compact topology on a fixed set $X$ some "dual" of a Hausdorff topology on $X$?
Here it is stated that a Hausdorff topology need not contain a minimal Hausdorff topology. If there is some duality these notions then we could also automatically deduce that a compact topology must not be contained in a maximal compact topology.

REPLY [15 votes]: Hausdorff is dual to discrete. Compact is dual to overt.
A space $X$ is Hausdorff if and only if the diagonal $\Delta_X = \{(x,x) \mid x \in X\}$ is closed in $X \times X$. A space $X$ is discrete if and only if $\Delta_X$ is open in $X \times X$.
Given a space $X$ let $\mathcal{O}(X)$ be its topology, seen as a topological space equipped with the Scott topology. Let $\Sigma = \{\bot, \top\}$ be the Sierpinski space.
Now, a space $X$ is compact if and only if the map $\forall_X: \mathcal{O}(X) \to \mathcal{O}(1)$, defined by
$$\forall_X (U) = \begin{cases}
\top & \text{if $U = X$} \\
\bot & \text{else}
\end{cases}$$
is continuous. A space is overt if and only if the map $\exists_X : \mathcal{O}(X) \to \mathcal{O}(1)$, defined by
$$\exists_X (U) = \begin{cases}
\top & \text{if $U \neq \emptyset$} \\
\bot & \text{else}
\end{cases}$$
is continuous. See Part II of Martin Escardo's Synthetic topology
of data types and classical spaces for a good tutorial on these ideas, or for a deep dive into the subject Paul Taylor's work on Abstract Stone Duality, and in particular his Foundations for Computable Topology.