Dimension of distributive lattices and universal spaces (Q1183643)

The main result is a theorem about the existence of universal spaces for Krull dimension. The key point is the factorization theorem for finite spaces. From some consequences for the dimension of lattices we mention the factorization theorem for distributive lattices. Factorization theorem for finite spaces: Let \(f: X\to Y\) be a continuous map between finite spaces. Then \(f\) factors through a finite space of dimension \(\leq n\) if and only if there is no chain in \(X\) of length \(n+1\) such that \(f\) is injective on it. Existence of universal spaces: There exists an \(n\)-dimensional space \(X_{w,n}\) of weight \(w\) such that any topological space of weight \(\leq w\) and dimension \(\leq n\) is homeomorphic to a subspace of \(X_{w,n}\). Factorization theorem of distributive lattices: Let \(f: A\to B\) be a lattice morphism. Then there exists a sublattice \(C\) of \(B\) satisfying the following conditions: (1) \(C\) contains the image of \(f\) and \(\dim C\leq\dim B\), (2) If \(A\) is finite, then so is \(C\). If \(A\) is infinite, then \(\text{card}C\leq\text{card}A\).