In recent work, Antiochos and coworkers argued that the boundary between the open and closed field regions on the Sun can be extremely complex with narrow corridors of open flux connecting seemingly disconnected coronal holes from the main polar holes, and that these corridors may be the sources of the slow solar wind. We examine, in detail, the topology of such magnetic configurations using an analytical source surface model that allows for analysis of the field with arbitrary resolution. Our analysis reveals three important new results: First, a coronal hole boundary can join stably to the separatrix boundary of a parasitic polarity region. Second, a single parasitic polarity region can produce multiple null points in the corona and, more important, separator lines connecting these points. It is known that such topologies are extremely favorable for magnetic reconnection, because they allow this process to occur over the entire length of the separators rather than being confined to a small region around the nulls. Finally, the coronal holes are not connected by an open-field corridor of finite width, but instead are linked by a singular line that coincides with the separatrix footprint of the parasitic polarity. We investigate how the topological features described above evolve in response to the motion of the parasitic polarity region. The implications of our results for the sources of the slow solar wind and for coronal and heliospheric observations are discussed.
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