Can a One Parameter Family of Metrics Be Metricizable?

[Haelfix]

Sorry about the horrible confusion in this post, I am befuddled and likely spouting gibberish.

Consider the general case of a one parameter family of metrics, where the parameter in question need not be finite (it can limit to infinity for instance). I am looking for a topology that is big enough to accommodate the entire family. (im not even sure how this should work, since I am used to thinking off one topology for one metric)

It seems to me in general, whatever this huge topology is, it need not and probably cannot be metricizable. Is this true, and can you think of any other constraints?

 

[Chris Hillman]

Hmm... lots of large topological spaces are metrizable, so don't give up hope. But without more context (maybe some examples of the kind of one parameter family of metrics you are looking at it?) I at least cannot suggest a candidate for a suitable topology.

It's nice to see an intriguing question, though

 

[George Jones]

I don't know enough to be any of help (hopefully Chris can), but I do know that Hawking used a similar idea in his formulation of the concept of stable causality. A spacetime (M,g) is stably causal if there isn't another Lorentz metric g' "close" to g such that (M,g') admits closed timlike curves.

Hawking defined "close" using a suitable topology on a collection of Lorentzian metrics on M. See pages 197/198 of Hawking and Ellis.