Does anybody want to give me a hint on problem 3?
H(R) is the set of all non-empty compact subsets of the reals (i.e. closed and bounded).
To show that the function is continuous I'm trying to use the fact that it would be equivalent to showing the pre-image of a closed subset of H(R) is closed in [0,1].
However, I don't know how to execute this.
Other urls found in this thread:
dropbox.com
en.wikipedia.org
twitter.com
what is the topology on H?
I guess it's the usual topology on R where open sets are open intervals with the Euclidean metric.
? how so? Isn't H a collection of compact sets?
Yeah, non-empty compact subsets of R.
so? what is the topology? show me an open set in H
My guess is that you get an open set in H by taking an open set in the reals, and then defining the set of all compact sets inside this open set to be open.
Don't know if this helps, if d([a,a+x],[a,a+y]) = d(x,y) then f_a is an isometry then it's continous
btw, which textbook are you using?
I'm not quite sure what an open set would look like but a closed set is something like [a,a]={a} or [a,a+x] for some x in (0,1].
Sorry for the confusion. I don't understand the question very well.