Topology of slices through the Sierpiński tetrahedron

We investigate slices of the Sierpiński tetrahedron from a topological viewpoint. For each $c\in [0,1]$, we study the Čech (co)homology group of the slice at height $c$. We show that the topology of the slice exhibits a sharp dichotomy. If $c$ is a dyadic rational, then the slice has finitely many connected components, infinite first Čech homology, and trivial higher homology. If $c$ is not a dyadic rational, then the slice is totally disconnected and all positive-degree Čech homology groups vanish.