I was asked during this talk whether the downset topology is always compatible with the lattice operations on a discrete lattice. I answered somewhat positively during the talk, but then I thought I remembered reading that this natural choice of a topology doesn't always work. I checked the smallest nontrivial example in this livestream ( ua-cam.com/video/pVoFfZAyXzk/v-deo.html ). I now realize that it's not hard to show from the definition that the operations for any lattice are continuous with respect to the downset topology, even for infinite lattices.
I was asked during this talk whether the downset topology is always compatible with the lattice operations on a discrete lattice. I answered somewhat positively during the talk, but then I thought I remembered reading that this natural choice of a topology doesn't always work. I checked the smallest nontrivial example in this livestream ( ua-cam.com/video/pVoFfZAyXzk/v-deo.html ). I now realize that it's not hard to show from the definition that the operations for any lattice are continuous with respect to the downset topology, even for infinite lattices.