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We investigate C-sets in almost zero-dimensional spaces, showing that closed
$\sigma $
C-sets are C-sets. As corollaries, we prove that every rim-
$\sigma $
-compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show that these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of
$($
Cantor set
$)\!\times \mathbb R$
is nowhere dense.
The aim was to evaluate under protein-limiting conditions the effect of different supplemental energy sources: fermentable NSP (fNSP), digestible starch (dStarch) and digestible unsaturated fat (dUFA), on marginal efficiency of fat deposition and distribution. A further aim was to determine whether the extra fat deposition from different energy sources, and its distribution in the body, depends on feeding level. A total of fifty-eight individually housed pigs (48 (sd 4) kg) were used in a 3 × 2 factorial design study, with three energy sources (0·2 MJ digestible energy (DE)/kg0·75 per d of fNSP, dStarch and dUFA added to a control diet) at two feeding levels. Ten pigs were slaughtered at 48 (sd 4) kg body weight and treatment pigs at 106 (sd 3) kg body weight. Bodies were dissected and the chemical composition of each body fraction was determined. The effect of energy sources on fat and protein deposition was expressed relative to the control treatments within both energy intake levels based on a total of thirty-two observations in six treatments, and these marginal differences were subsequently treated as dependent variables. Results showed that preferential deposition of the supplemental energy intake in various fat depots did not depend on the energy source, and the extra fat deposition was similar at each feeding level. The marginal energetic transformation (energy retention; ER) of fNSP, dStarch and dUFA for fat retention (ERfat:DE) was 44, 52 and 49 % (P>0·05), respectively. Feeding level affected fat distribution, but source of energy did not change the relative partitioning of fat deposition. The present results do not support values of energetic efficiencies currently used in net energy-based systems.
The space now known as complete Erdős space${{\mathfrak{E}}_{\text{c}}}$ was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space ${{\ell }^{2}}$ consisting of all vectors such that every coordinate is in the convergent sequence $\left\{ 0 \right\}\cup \left\{ 1/n:n\in \mathbb{N}\ \right\}$. In a solution to a problem posed by Lex $G$. Oversteegen we present simple and useful topological characterizations of ${{\mathfrak{E}}_{\text{c}}}$. As an application we determine the class of factors of ${{\mathfrak{E}}_{\text{c}}}$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces ${{\ell }^{p}}$ according to the ‘Erdős method’ are homeomorphic to ${{\mathfrak{E}}_{\text{c}}}$. A novel application states that if $I$ is a Polishable ${{F}_{\sigma }}$-ideal on $\omega $, then $I$ with the Polish topology is homeomorphic to either $\mathbb{Z}$, the Cantor set ${{2}^{\omega }},\,\mathbb{Z}\,\times \,{{2}^{\omega }}$, or ${{\mathfrak{E}}_{\text{c}}}$. This last result answers a question that was asked by Stevo Todorčević.
In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\mathbb{R}$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\mathbb{Q}$ onto itself is homeomorphic to the infinite power of $\mathbb{Q}$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\mathbb{Q}$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ${{\mathbb{R}}^{n}}$ for $n\ge 2$, by linking the groups in question with Erdős space.
In 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $\ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $\mathcal{E}$ of the Banach spaces $\ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $\mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $\mathcal{E}$ is closed in $\ell^p$, then it is homeomorphic to complete Erdős space if and only if $\dim\mathcal{E}>0$ and every $E_n$ is zero dimensional.
It is proved that the countably infinite power of complete Erdős space $\Ec$ is not homeomorphic to $\Ec$. The method by which this result is obtained consists of showing that $\Ec$ does not contain arbitrarily small closed subsets that are one-dimensional at every point. This observation also produces solutions to several problems that were posed by Aarts, Kawamura, Oversteegen and Tymchatyn. In addition, we show that the original (rational) Erdős space does contain arbitrarily small closed sets that are one-dimensional at every point.
A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in
the plane is presented. It was claimed that this result can be generalized for sets that meet every line in
either one point or two points. No proof of this assertion is known, however. The main results in this
paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set
that meets every line in the plane in at least one but at most two points must be zero-dimensional, and
that if it is σ-compact then it must be a nowhere dense Gδ-set in the plane. Generalizations for similar
sets in higher-dimensional Euclidean spaces are also presented.
A shadow of a subset A of ℝn is the image of A
under a projection onto a hyperplane. Let C be a closed nonconvex set in ℝn
such that the closures of all its shadows are convex. If, moreover, there are n
independent directions such that the closures of the shadows of C in those directions are proper subsets
of the respective hyperplanes then it is shown that C contains a copy of ℝn−2.
Also for every closed convex set B ‘minimal imitations’ C of B are constructed,
that is, closed subsets C of B that have the
same shadows as B and that are minimal with respect to dimension.
It is shown that under $\text{ZFC}$ almost all planar compacta that meet every line in at most two points are subsets of sets that meet every line in exactly two points. This result was previously obtained by the author jointly with K. Kunen and J. vanMill under the assumption that Martin’s Axiom is valid.
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