Point -like upper semi-continuous decompositions of $S^{3}$
John L. Bailey · Illinois Journal of Mathematics · 1969
In this paper it is shown that point-like upper semi-continuous decompo- sitions of S which satisfy certain conditions on the distribution of their non- degenerate elements are topologically equivalent to S3.In [3], J. F. Wardwell obtained similar results for arbitrary compact metric spaces but stronger hy- potheses were necessary.The proof of Theorem 1 of [2] by R. H. Bing is used to obtain a stronger result for S3.This proof shows that if for each arbitrary open set U containing the nondegenerate elements of a point-like upper semi- continuous countable decomposition G, if for each e > 0 there exists a homeo- morphism h of E onto E which shrinks each element of G into a set of diameter less than and which is fixed on E U, then E3/G E3.It is easy to see the proof also applies to S.A point-like set in S is one whose complement is topologically equivalent to the complement of a point.For a decomposition G we define H0 (G) g e G g is nondegenerate},and define recursively Hk(G) Ig e Ho(G) g n lim sup H(G)This motivates a generalization for any ordinal number H,(G) {g e Ho(G) g n lim sup H(G) (In the following the symbol "=" will also mean "is homeomorphic to".It should be clear from the context when this is meant and when strict equality is meant.Converting his results to the notation which I will use, J. F. Wardwell proved in [3]:LEMMA.If G is an upper semi-continuous decomposition of a compact metric space M into point-like sets and there exists a positive integer ]c such that Hk (G) , then M/G M.THEOnEM.If G is an upper semi-continuous decomposition of a compact metric space M into point-lilce sets, if o (lira sup H(G) is zero-dimensional, and iffor some countable ordinal a, H,(G) 0; then M/G M.In Theorem 2 of this paper the above theorem is proved for S with weakened hypotheses.Theorem 2 is applied in Theorem 3 to show how "bad" a point-