621 research outputs found
Coloring Cantor sets and resolvability of pseudocompact spaces
Let us denote by the statement that , i.e. the Baire space of weight , has a coloring
with colors such that every homeomorphic copy of the Cantor set
in picks up all the colors.
We call a space {\em -regular} if it is Hausdorff and for every
non-empty open set in there is a non-empty open set such that
. We recall that a space is called {\em feebly
compact} if every locally finite collection of open sets in is finite. A
Tychonov space is pseudocompact iff it is feebly compact.
The main result of this paper is the following.
Theorem. Let be a crowded feebly compact -regular space and be
a fixed (finite or infinite) cardinal. If holds for all
then is -resolvable, i.e. contains
pairwise disjoint dense subsets. (Here is the smallest
cardinal such that does not contain many pairwise
disjoint open sets.)
This significantly improves earlier results of van Mill , resp.
Ortiz-Castillo and Tomita.Comment: 8 page
Pinning Down versus Density
The pinning down number of a topological space is the smallest
cardinal such that for any neighborhood assignment
there is a set with for all . Clearly, c.
Here we prove that the following statements are equivalent:
(1) for each cardinal ;
(2) for each Hausdorff space ;
(3) for each 0-dimensional Hausdorff space .
This answers two questions of Banakh and Ravsky.
The dispersion character of a space is the smallest
cardinality of a non-empty open subset of . We also show that if
then has an open subspace with and
, moreover the following three statements are equiconsistent:
(i) There is a singular cardinal with , i.e.
Shelah's Strong Hypothesis fails;
(ii) there is a 0-dimensional Hausdorff space such that
is a regular cardinal and ;
(iii) there is a topological space such that is a regular
cardinal and .
We also prove that
for any locally compact Hausdorff space ;
for every Hausdorff space we have and
implies ;
for every regular space we have and moreover implies
Anti-Urysohn spaces
All spaces are assumed to be infinite Hausdorff spaces. We call a space
"anti-Urysohn" AU in short iff any two non-emty regular closed sets in it
intersect. We prove that
for every infinite cardinal there is a space of size
in which fewer than many non-empty regular closed
sets always intersect;
there is a locally countable AU space of size iff .
A space with at least two non-isolated points is called "strongly
anti-Urysohn" SAU in short iff any two infinite closed sets in it
intersect. We prove that
if is any SAU space then ;
if then there is a separable, crowded,
locally countable, SAU space of cardinality ; \item if Cohen reals are added to any ground model then in the extension there
are SAU spaces of size for all ;
if GCH holds and are uncountable regular
cardinals then in some CCC generic extension we have ,
, and for every cardinal there is an SAU space of cardinality .
The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality can exist remain open
Random transverse-field Ising chain with long-range interactions
We study the low-energy properties of the long-range random transverse-field
Ising chain with ferromagnetic interactions decaying as a power alpha of the
distance. Using variants of the strong-disorder renormalization group method,
the critical behavior is found to be controlled by a strong-disorder fixed
point with a finite dynamical exponent z_c=alpha. Approaching the critical
point, the correlation length diverges exponentially. In the critical point,
the magnetization shows an alpha-independent logarithmic finite-size scaling
and the entanglement entropy satisfies the area law. These observations are
argued to hold for other systems with long-range interactions, even in higher
dimensions.Comment: 6 pages, 4 figure
STRONG COLORINGS YIELD kappa-BOUNDED SPACES WITH DISCRETELY UNTOUCHABLE POINTS
It is well known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklossy and the first author, we show that this statement fails for countably compact regular spaces, and even for omega-bounded regular spaces. In fact, there are kappa-bounded counterexamples for every infinite cardinal kappa. The proof makes essential use of the so-called strong colorings that were invented by the second author
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