The connectedness of the collection of arc cluster sets

Abstract:

Let $f$ be a continuous complex-valued function defined on the unit disk and let $p$ be a boundary point of the disk. A very natural topology on the collection of all arc cluster sets of $f$ at the point $p$ has been investigated by Belna and Lappan [1] who proved that this collection is a compact set under certain suitable conditions. It is proved here that this collection is an arcwise connected set under the topology in question, but is not in general locally arcwise connected or even locally connected. It is also shown by example that it is generally not possible to map the real line onto the collection of arc cluster sets at $p$ in a continuous manner.