Concerning upper semi-continuous collections of continua

Abstract

f If M is a point set and P is a point, then by I (PM) is meant the lower bound of the distances from P to all the different points of M. If M and N are two point sets, then by l(MN) is meant the lower bound of the values [l(PN)} for all points P of M, while by u(MN) is meant the upper bound of these values for all points P of M. It is to be observed that u{MN) may be different from u(NM).

Planar finitely Suslinian compacta

“…Note also that p −1 (z) is a nonseparating plane continuum for each z. By a theorem of Moore [10], C/ is homeomorphic to the plane. By construction the set Z is unshielded in the plane C/ , and the fact that Z is FS follows from Lemma 1.3.…”

Tame subsets of spheres inE3

“…From (6) the decomposition G of S whose only nondegenerate elements are the A t 9 s is upper semicontinuous. It follows from (4) and [18] that the decomposition space is a 2-sphere. Since there are only a countable number of nondegenerate elements in G, the image of these nondegenerate elements forms a countable point set in the decomposition space.…”

“…Notice that C = C, + C 2 , and C = C/ + C/. If D is in C 2 , then h λ {D) intersects F and h γ {D) -J^ Int ^ lies in Int R. We will first show how to obtain h z (D) for the disks in C u Using (*, F, ΈxtR), we let S' be a polyhedral 2-sphere and we let h r be a homeomorphism of R onto £' such that (18) h' moves no point as much as ε 3 , (19) S' contains a finite collection of disjoint ε 3 -disks such that S' minus these disks lies in Exti2,…”

“…The only difference is that we use (*, F, Int 22) to obtain a polyhedral 2-sphere S", homeomorphically within ε 3 of 22, which "lies almost in Int 22 and misses F" (that is, S" satisfies conditions similar to (18), (19), (20), and (21)). For each disk D in C 2 , h x (D) will lie "almost" in Int S", so we can "pull hJJD) and S" apart" just as we "pulled hJJ)) and S' apart" in the preceding paragraph to obtain a polyhedral disk h 2 (D) in IntS".…”

Cell-like 0-dimensional decompositions of 𝐸³

“…Let A/: E3 -» E3 be a homeomorphism such that h'\E3 -2 = id and A/(2 n A') c {M u (U*_iJF,)) where {ir,}î=1 is a collection of disjoint tame disks in Int 2, Ft n M = Bd Ft n M = an arc F" (Bd F, -Ft) n A/04') = 0, and (Af n A/(y4')) u U kt=XF, is a finite graph T on M. By Moore's theorems [5], there is a homeomorphism of M to itself fixed on Bd M which takes T into the set U7>, Vj u (degenerate elements of G). Let A": E3 -» 2s3 be a homeomorphism which extends the above homeomorphism of Af and has the property that h"\E3 -2 = id.…”