3

Does there exist a field with a topology such that addition and multiplication are continuous, but division is not (ignoring dividing by 0, i.e. it's not even continuous-where-defined)?

Anonymous
  • 31
  • 1
    Interesting question. What I do know is this: if the topology is complete metric, then inverse is continuous. – GEdgar Sep 15 '15 at 12:30
  • What I also know is that for every topology there is a smaller topology making 1/x continous (take the induced topology in the projective line). So the question is equivalent to 'is every topological field homeomorphic to its image in its own projective line?' – Anonymous Sep 15 '15 at 12:36
  • 1
    Possible duplicate of group of units in a topological ring – Jonathan Gleason Jan 04 '17 at 03:34
  • 1
    Yes. See http://math.stackexchange.com/questions/1393303/group-of-units-in-a-topological-ring.

    You can find this in Warner's Topological Fields on pg. 113. In brief, you take the rationals and equip them with a topology in which a neighborhood base of $0$ is given by $n\mathbb{Z}$ for $n\in \mathbb{Z}^+$.

     – Jonathan Gleason Jan 04 '17 at 03:37

0 Answers0