A paratoplogical group is a group equipped with a topology in which multiplication, but not necessarily inversion, is continuous. An example is the Sorgenfry line under addition. In the present paper, we propose a reasonable point-free version of this notion – a paralocailc group. We provide several examples and non-examples, and we list several unsolved problems. Among our examples, we show that addition does not induce the structure of a paralocailc group on the frame of opens of the Sorgenfrey line. This is related to the non-spatiality of the coproduct the Sorgenfry topology with itself.
Publication Source (Journal or Book title)
Topology and its Applications
Madden, J., & Mugochi, M. (2019). Paralocalic groups. Topology and its Applications, 259, 275-282. https://doi.org/10.1016/j.topol.2019.02.035