Rationals as a non-trivial complete convergence group
Petr Šimon · Czechoslovak Mathematical Journal · 1996
It is a well-known fact that convergence spaces, despite of their seemingly close relation to first countable topological spaces, do not admit a reasonable notion of completeness.This obstacle may be overcome by imposing more structure on the underlying set: As proved by J. Novak, there is a sound notion of a Cauchy filter on a convergence group, and every convergence group has a completion [N].However, numerous papers pointed out that even in the most elementary setting, namely (Q, +), things may go weird (see e.g.[Fl,F2]).Since-up to the author's knowledgenobody has paid attention to those group convergences on rationals which are strictly finer than the usual metric one, we want to show that it may even happen that Q is complete in such a case.(Another example of this kind may be found in [DFZ], with the convergence coarser than the metric convergence and the induced closure antiHausdorff.)We do not consider the result just another bizzare example, because we feel that it provides some information on the complexity of those sequences of rationals which converge to an irrational number.However, it is also true that for every irrational number x there is a group convergence ^ on the rationals such that for its categorical completion (Q, ^f) one has x G Q and still U \ Q ^ 0. This is a special case of our Theorem 2, where we characterize compact subsets X of U such that for some group convergence ^ on Q, finer than the usual metric one, QuICQCR.For the reader's convenience, let us recall the basic notions from the theory of convergence groups.Let X be a set.A subset ^ C U X x X is called a convergence on X provided the following holds:(S) for each x G X we have ((x: n e u),x) G ^, where (x: n e co) denotes the constant sequence with value x; Supported by CNR grant no.218.1495 and by GAUK 350.