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First, a brief introduction of the context. A usc decomposition of a metric space $X$ is a collection $G$ of compact sets which vary semicontinuously. I will be interested in cellular decompositions of $\mathbb{R}^n$ (that is, the elements of $G$ are decreasing intersections of closed balls).

In 1967, Jones proved that it is not possible to fill $\mathbb{R}^n$ with disjoint arcs. This came after Roberts had proven that the plane can be decomposed by cellular sets none of which is a point (or equivalently, there is a lower bound on the diameter of the sets).

After some time searching the web, etc, I've found out that there has been a lot of progress towards the understanding of cellular decompositions of manifolds (see Daverman's book) but some kind of questions have been somewhat not treated.

Is there some progress towards the understanding of cellular decompositions of $\mathbb{R}^n$ with all of the elements of diameter bounded from below? In particular, I am interested in the following questions:

Q1) What is the possible topology of the sets constituting such a decomposition? Mainly, which kind of possible topologies are compatible, etc.

In general (at least in dimension 2) such a decomposition is equivalent to the existence of a surjective continuous map $f:\mathbb{R}^n \to \mathbb{R}^n$ such that the preimage of every point is an element of $G$ (in particular, compact connected and cellular).

So, assuming $f:\mathbb{R}^n \to \mathbb{R}^n$ is a surjective continuous map such that the preimage of each point is cellular and of diameter bigger than $0$

Q2) What kind of preimages may arcs have under $f$? Or disks? What is the topology of this sets?

And the question I am most interested in is:

Q3) Consider an open set $U\in \mathbb{R}^n$ such that $f(U)$ is open: Does it hold that $U$ contains a fiber of $f$ (that is, a set of the form $f^{-1}(x)$)?

Answers in dimension $2$ are already very interesting to me. Other related references will be appreciated.

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Interesting that space cannot be filled with arcs, because it can be filled with disjoint loops (or "hoops"): . – Joseph O'Rourke Oct 19 '10 at 14:35
Thanks! However, I am mostly interested in decompositions in sets which are cellular, this imposes more restrictions. – rpotrie Oct 21 '10 at 11:09

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