The Bolzano property and the cube-like complexes (Q2403349)

The author introduces an internal characterization of the Bolzano property as follows: a topological space \(X\) is said to have the \(n\)-dimensional Bolzano property if there exists a family \(\{(A_i,B_i):i=1,\dots,n\}\) of pairs of disjoint closed subsets of \(X\) such that for every family \(\{(H_i^-,H_i^+):i=1,\dots,n\}\) of closed sets such that for each \(0<i\leq n\), \(A_i\subset H_i^-\), \(B_i\subset H_i^+\), and \(H_i^-\cup H_i^+=X\), we have \(\bigcap_{i=1}^n(H_i^-\cap H_i^+)\neq\emptyset\). Two proofs of the fact that \(n\)-cube-like polyhedra in \(\mathbb R^m\) have the \(n\)-dimensional Bolzano property are presented, implying that the Poincaré-Miranda theorem is valid for such complexes. Further, stability of the Bolzano property under inverse limits is investigated, as well as its characterization for locally connected spaces.