The lattice of Knaster continua (Q1868025)

Knaster continua are special subcontinua of the Hilbert cube \(\mathbb Q\) and are defined as inverse limits of inverse sequences of the closed unit intervals. It is shown that the set \(\mathbb K\) of homeomorphism classes of Knaster continua is a lattice with respect to a partial order defined by open mappings between these continua. For Knaster continua \(X\) and \(Y\), \(X\leq Y\) means that there is an open mapping \(f:X\to Y\), and \([X]\leq [Y]\) iff \(X\leq Y\). A description of this lattice is given, its structure is investigated and some questions concerning open mappings on Knaster continua are posed.