Spaces of Compact Open Subgroups and Open Morphisms

For a morphism whose domain is a totally disconnected locally compact group, the openness test of Cowling, Hofmann and Morris [‘Open mappings of locally compact groups’, J. Group Theory 27 (6) (2024), 1143–1149] can be stated in terms of compact open subgroups. We use this formulation to study spaces of compact open subgroups. A surjective open morphism with compact kernel induces a quotient map between such spaces, and the inverse-image map identifies the target with a closed retract of the source. Conversely, for a surjective open morphism whose target has a compact open subgroup, compactness of the kernel is forced by the requirement that inverse images of compact open subgroups remain compact.