986 research outputs found
Countable dense homogeneity in powers of zero-dimensional definable spaces
We show that, for a coanalytic subspace of , the countable
dense homogeneity of is equivalent to being Polish. This
strengthens a result of Hru\v{s}\'ak and Zamora Avil\'es. Then, inspired by
results of Hern\'andez-Guti\'errez, Hru\v{s}\'ak and van Mill, using a
technique of Medvedev, we construct a non-Polish subspace of
such that is countable dense homogeneous. This gives the first
answer to a question of Hru\v{s}\'ak and Zamora Avil\'es.
Furthermore, since our example is consistently analytic, the equivalence result
mentioned above is sharp. Our results also answer a question of Medini and
Milovich. Finally, we show that if every countable subset of a zero-dimensional
separable metrizable space is included in a Polish subspace of then
is countable dense homogeneous.Comment: 14 page
A non-CLP-compact product space whose finite subproducts are CLP-compact
We construct a family of Hausdorff spaces such that every finite product of
spaces in the family (possibly with repetitions) is CLP-compact, while the
product of all spaces in the family is non-CLP-compact. Our example will yield
a single Hausdorff space such that every finite power of is
CLP-compact, while no infinite power of is CLP-compact. This answers a
question of Stepr\={a}ns and \v{S}ostak.Comment: 8 page
Products and countable dense homogeneity
Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we
construct a zero-dimensional separable metrizable space such that is
countable dense homogeneous while is not. It follows from results of
Hru\v{s}\'ak and Zamora Avil\'es that such a space cannot be Borel.
Furthermore, can be made homogeneous and completely Baire as well.Comment: 7 page
Products and h-homogeneity
Building on work of Terada, we prove that h-homogeneity is productive in the
class of zero-dimensional spaces. Then, by generalizing a result of Motorov, we
show that for every non-empty zero-dimensional space there exists a
non-empty zero-dimensional space such that is h-homogeneous.
Also, we simultaneously generalize results of Motorov and Terada by showing
that if is a space such that the isolated points are dense then
is h-homogeneous for every infinite cardinal . Finally, we show that a
question of Terada (whether is h-homogeneous for every
zero-dimensional first-countable ) is equivalent to a question of Motorov
(whether such an infinite power is always divisible by 2) and give some partial
answers.Comment: 10 page
Productively Lindel\"of spaces of countable tightness
Michael asked whether every productively Lindel\"of space is powerfully
Lindel\"of. Building of work of Alster and De la Vega, assuming the Continuum
Hypothesis, we show that every productively Lindel\"of space of countable
tightness is powerfully Lindel\"of. This strengthens a result of Tall and
Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and
Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a
productively Lindel\"of space is powerfully Lindel\"of if every open cover
of admits a point-continuum refinement consisting of basic open
sets. This strengthens a result of Burton and Tall. Finally, we show that
separation axioms are not relevant to Michael's question: if there exists a
counterexample (possibly not even ), then there exists a regular
(actually, zero-dimensional) counterexample.Comment: 7 page
Atomic Scale Fractal Dimensionality in Proteins
The soft condensed matter of biological organisms exhibits atomic motions
whose properties depend strongly on temperature and hydration conditions. Due
to the superposition of rapidly fluctuating alternative motions at both very
low temperatures (quantum effects) and very high temperatures (classical
Brownian motion regime), the dimension of an atomic ``path'' is in reality
different from unity. In the intermediate temperature regime and under
environmental conditions which sustain active biological functions, the fractal
dimension of the sets upon which atoms reside is an open question. Measured
values of the fractal dimension of the sets on which the Hydrogen atoms reside
within the Azurin protein macromolecule are reported. The distribution of
proton positions was measured employing thermal neutron elastic scattering from
Azurin protein targets. As the temperature was raised from low to intermediate
values, a previously known and biologically relevant dynamical transition was
verified for the Azurin protein only under hydrated conditions. The measured
fractal dimension of the geometrical sets on which protons reside in the
biologically relevant temperature regime is given by . The
relationship between fractal dimensionality and biological function is
qualitatively discussed.Comment: ReVTeX4 format with 5 *.eps figure
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