On open mappings of locally connected continua onto arcs (Q2705776)

The authors study the question which continua admit an open retraction onto an arc. They present a proof of the following result of Prajs: If \(X\) is a locally connected continuum then \(X\) admits an open retraction onto an arc iff \([0,1]\) is an open image of \(X\) iff there are two disjoint nowhere dense closed subsets \(A\) and \(B\) of \(X\) with \(C(A,B)=X\). The last condition means that for every \(p\in X\smallsetminus (A\cup B)\) there is an arc \(L\) in \(X\) with endpoints \(a\) and \(b\) such that \(p\in L\), \(L\cap A=\{a\}\) and \(L\cap B=\{b\}\). The authors add several new equivalent statements to this list, such as: there exists a graph \(G\) such that \(G\) is an open image of \(X\) iff for each \(A\)-set \(Y\) in \(X\) and for each point \(p\in X\) the sets \( X\smallsetminus (\text{bd} Y)^d\) and \(X\smallsetminus \{p\}\) have finitely many components only, etc. The authors also pose the following interesting problem: Which continua admit an open map onto an arc? Several useful observations and comments are made.