Topology is hard

what is the topology on H?

>>8539377
I guess it's the usual topology on R where open sets are open intervals with the Euclidean metric.

>>8539424
? how so? Isn't H a collection of compact sets?

>>8539440
Yeah, non-empty compact subsets of R.

>>8539465
so? what is the topology? show me an open set in H

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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.

>>8539369
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

>>8539369
btw, which textbook are you using?

>>8539481
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.

>>8539518

No textbook. We're using some notes our professor compiled.

Here's the chapter this problem is from. https://www.dropbox.com/s/d0ai3iqzicz4kc9/hyperspace%202-4.pdf?dl=0

>>8539524
After reading the notes I'm pretty sure the answer is >>8539502

>>8539377
It's the Vietoris topology.

For every [math]a \in \mathbb{R} [/math] and [math] x \in [0,1] [/math] we have
[eqn] \lim_{y \to 0} h(f_a(x+y) , f_a(x)) = \lim_{y \to 0} y = 0 [/eqn]
Therefore [math]f_a [/math] is continuous.

>>8539369
Continuity is a topologic property, if we don't know the topology, we can't help you. (what is h in problem 2?)

>>8539570
It's the Hausdorff distance
https://en.wikipedia.org/wiki/Hausdorff_distance

The topology is generated by this metric.

Alternatively in this metric every open set of the functions range takes the form of all intervals [a,a+x] where x is in some (a,b) contained in [0,1], hence every open set of the functions range has an open inverse image.

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Thanks guys, you've helped out a lot.

If anyone would like provide another hint for a different problem it would be much appreciated.

I'm trying to prove that the collection of sets in Definition 1 is a basis for some topology (Vietoris?).

>>8539570
Ok so an open ball of center fa(x) and radius r would be the set of all compacts of R contained in ]a-r , a+x+r[ .
Are you ok with that, because you seem confused about what an open set of H looks like ?

>>8539650
Okay, that makes sense. By ]a-r , a+x+r[ do you mean the complement of [a-r,a+x+r]?

>>8539650
Thats not quite correct. An open ball of center fx(x) and radius r would be the set of all sets of the form [a, a + x + h], where |h| < r. The left endpoint must be "a" for interval to be in the range of f.

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Not OP but can someone clear up the confusion of a topology noob

How is [0, 1] a topological space when paired with the usual Euclidean topology? The book hasn't mentioned subset topologies at all. Does the restriction of the Euclidean metric to [0, 1] somehow induce a subset topology? Otherwise how can there be an open interval around 0 or 1?

>>8539665
He means (a-r, a+x+r).

>>8539697
He's right, the metric is the Hausdorff metric.

>>8539753
An open set in the subspace doesn't need to be open in the entire space. [0, 1/2) is open in [0, 1], for example.

>>8539759
>An open set in the subspace doesn't need to be open in the entire space. [0, 1/2) is open in [0, 1], for example.
Alright so I'm guessing that the open ball function in (R, d) is different from the one in [math](I, d\vert_I)[/math]? That would make sense.