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Finite connected partially ordered sets are in bijective correspondence to connected finite topological spaces that satisfy T_0, see for example the Wikipedia article Finite topological space. Here ...

Here's a fairly easy fact from point-set topology that I'm having trouble finding a reference for. Say $X$ is a total order satisfying the least-upper bound property, and $S$ is a closed subset of it....

Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...

A continuous lattice may be defined as a complete lattice in which arbitrary meets distribute over directed joins. A continuous lattice is naturally regarded as an algebraic structure where the ...