Confluent mappings and arc Kelley continua (Q1011049)

A metric continuum \(X\) is called a Kelley continuum provided that for each point \(p\), each continuum \(K\) in \(X\) containing \(p\) and each sequence of \(p_n\) converging to \(p\), there exists a sequence of subcontinua \(K_n\) containing \(p_n\) that converges to \(K\). A Kelley continuum \(X\) is called an arc continuum provided that each subcontinuum \(K\) of \(X\) containing \(p\) can be approximated by arcwise connected continua containing \(p\). A continuum homeomorphic to the inverse limit of locally connected continua with confluent bonding maps is said to be confluently \textit{LC}-representable. The authors study the close relationships between the arc Kelley continua and confluent mappings. The main result says that if a continuum \(X\) admits, for each \(\epsilon > 0\), a confluent \(\epsilon\)-mapping onto a(n) (arc) Kelley continuum, then \(X\) itself is a(n) (arc) Kelley continuum. In particular, each confluently \textit{LC}-representable continuum is arc Kelley. The authors study conditions under which the different hyperspaces of a continuum \(X\) are \textit{LC}-representable. They also offer a number of interesting open problems.