Strongly freely decomposable mappings, almost monotone mappings and \(i\)-unicoherent continua (Q2315331)

In this paper, a continuum is a nonempty compact connected metric space. A mapping \(f:X\to Y\) between continua is said to be strongly freely decomposable if whenever \(C\) and \(D\) are proper subcontinua of \(Y\) such that \(Y=C\cup D\), then \(f^{-1}(C)\) is connected. One says that \(f\) is almost monotone provided that \(f^{-1}(Q)\) is connected for each subcontinuum \(Q\) of \(Y\) with nonempty interior. Every almost monotone mapping is freely decomposable. In the paper, the authors study conditions and particular cases where the classes are equal. A new concept is introduced in Section 3, and some facts about this are developed. Definition 3.2. A continuum \(X\) is said to be \(i\)-unicoherent if whenever \(A\) and \(B\) are subcontinua of \(X\), \(X=A\cup B\), and \(\mathrm{int}_X(A\cap B)\neq\emptyset\), then \(A\cap B\) is connected. Although every unicoherent continuum is \(i\)-unicoherent, the converse is not true by dint of the authors' Example 3.3. In Section 4, examples of \(i\)-unicoherent continua are given. In Section 5, the authors show (Theorems 5.1 and 5.2) that the image of an \(i\)-unicoherent continuum under either a quasi-monotone or strongly freely decomposable mapping is also \(i\)-unicoherent (but this is not stated correctly in the last paragraph of Section 1). They study hereditarily \(i\)-unicoherent continua in Section 6.