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Register a new userCharacterizing indecomposable plane continua from their complements
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We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum.
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If $f:[a,b]to mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:mathbb{C}tomathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $Xto D, Xsubset D$ and for positively oriented maps $f:Xto mathbb{C}, Xsubset mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:mathbb{C}tomathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $ain mathrm{Int}(X)$ or $f$ exhibits rotation at $a$.
By the {em Suslinian number} $Sln(X)$ of a continuum $X$ we understand the smallest cardinal number $kappa$ such that $X$ contains no disjoint family $C$ of non-degenerate subcontinua of size $|C|gekappa$. For a compact space $X$, $Sln(X)$ is the smallest Suslinian number of a continuum which contains a homeomorphic copy of $X$. Our principal result asserts that each compact space $X$ has weight $leSln(X)^+$ and is the limit of an inverse well-ordered spectrum of length $le Sln(X)^+$, consisting of compacta with weight $leSln(X)$ and monotone bonding maps. Moreover, $w(X)leSln(X)$ if no $Sln(X)^+$-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of cite{DNTTT1}. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If $X$ is a continuum with $Sln(X)<2^{aleph_0}$, then $X$ is 1-dimensional, has rim-weight $leSln(X)$ and weight $w(X)geSln(X)$. Our main tool is the inequality $w(X)leSln(X)cdot w(f(X))$ holding for any light map $f:Xto Y$.
We obtain several game characterizations of Baire 1 functions between Polish spaces X, Y which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using similar ideas, we give game characterizations of Baire measurable and Lebesgue measurable functions.
In this paper we present proofs of basic results, including those developed so far by H. Bell, for the plane fixed point problem. Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. We develop a prime end theory through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum $X$. We define the concept of an {em outchannel} for a fixed point free map which carries the boundary of $X$ minimally into itself and prove that such a map has a emph{unique} outchannel, and that outchannel must have variation $=-1$. We also extend Bells linchpin theorem for a foliation of a simply connected domain, by closed convex subsets, to arbitrary domains in the sphere. We introduce the notion of an oriented map of the plane. We show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. We obtain a fixed point theorem for positively oriented, perfect maps of the plane. This generalizes results announced by Bell in 1982 (see also cite{akis99}). It follows that if $X$ is invariant under an oriented map $f$, then $f$ has a point of period at most two in $X$.
In his seminal work cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if $H$ is a compact subgroup of a locally compact group $G$ and $S$ is a small (in the sense of Palais) $H$-slice in a proper $G$-space, then the action map $Gtimes Sto G(S)$ is open. This is applied to prove that the slicing map $f_S:G(S)to G/H$ is continuos and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach-Mazur compacta.
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