Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/\_{\mathcal D}$ is homeomorphic to $\mathbb S^n$ if $\mathcal D$ is "shrinkable", and there are many conditions on $\mathcal D$ that ensure this. (For example, there are numerous such conditions in Daverman's book "Decompositions of Manifolds".)


However, the statements I've been able to find assume, in one way or another, that $\mathcal D'$ is countable.


Is there some set of conditions on an upper semicontinuous decomposition $\mathcal D$ of $\mathbb S^n$ that ensures that the decomposition space $^{\mathbb S^n}/\_{\mathcal D}$ is homeomorphic to $\mathbb S^n$ (e.g. by ensuring that $\mathcal D$ is shrinkable), but that does not require the set $\mathcal D'$ of non-singletons in $\mathcal D$ to be countable?


(In the situation I have in mind, $\cup\_{D\in\mathcal D'}D$ has dense, full-measure complement, each $D\in\mathcal D'$ is homeormorphic to a closed ball in $\mathbb E^k$ for some $k<n$, and $^{\mathbb S^n}/\_{\mathcal D}$ is compact, Hausdorff, and second-countable. Moreover, the quotient $\mathbb S^n\to^{\mathbb S^n}/\_{\mathcal D}$ has a very nice description. However, my $\mathcal D'$ is definitely uncountable.)



Shrinkable decompositions with uncountably many non-degenerate elements?
Bing's Dogbone Space stems from a decomposition of $S^3$ unto points
and an uncountable collection of tame arcs. The decomposition space is not a manifold,
so the decomposition is non-shrinkable. W. T. Eaton (Proc. Amer. Math. Soc. 39
(1973), 379--387) presented a higher dimensional analog of the Dogbone Space, involving a decomposition of $S^n$, $n>3$, into points and tame arcs that yields a nonmanifold.


Positive results in the general setting are nowhere near as rich as when the set of nondegenrate elements is countable. The best of them is Edwards's Cell-kike Approximation Theorem, which promises that for $n>4$ a cell-like decomposition of an $n$-manifold $M$ is shrinkable if the quotient space is finite dimensional and satisfies the Disjoint Disks Property.


