5

Does there exist a fruitful notion of "moduli space of topologies"?

For example, is it possible to define useful/natural topologies on the set of topologies on a given set $A$? When does it make sense to talk about the convergence of a sequence of topologies? Are there other interesting structures one can define on this space besides the topological?

If the set of all topologies over a given set is too unwieldy, are there useful restrictions that allow a more fruitful study?

How about topologies on distinct sets? For example, suppose $A_i$, $i=1,2,\ldots$ is a sequence of sets and $\tau_i$ is a topology on $A_i$. Is there a useful notion of convergence in such a setting?

merle
  • 63
  • 2
    It seems at first glance that the space of all topologies on a given set ought to be a complete lattice, as an arbitratry intersection of topologies is again a topology, and the topology generated by a union of topologies is the smallest topology greater than all of them. You can probably use this lattice structure to define a topology on the set of topologies. – Olivier Bégassat Aug 22 '13 at 19:48

1 Answers1

2

The set of all topologies along with the "coarser" partial relation indeed forms a complete bounded lattice. The maximum element is the discrete topology, the minimum element is the trivial topology. The "coarser" relation is partial, so you cannot put an order topology on it. For a related question, see here: Topology on the set of partitions

Community
  • 1
exk
  • 919
  • 5
  • 12