Lebesgue covering theorem

Statement
Suppose $$X$$ is a fact about::compact polyhedron, i.e., it is the underlying topological space, or geometric realization, of a fact about::finite simplicial complex which we denote by $$K$$. Then, the dimension of $$K$$ (defined as (size of largest face of $$K$$) minus one) equals the fact about::covering dimension of $$X$$.