


General Topology and Its Applications
Org: A. Karassev (Nipissing), M. Tuncali (Nipissing), E.D. Tymchatyn (Saskatoon) and V. Valov (Nipissing) [PDF]
 TARAS BANAKH, Nipissing University, North Bay, Canada and Lviv National
University, Lviv, Ukraine
Selection Principles and Games on Multicovered Spaces
[PDF] 
We shall discuss some Selection Principles and related games on
multicovered spaces. A multicovered space is a pair
(X,m) consisting of a set X and a family m of covers of X.
The category of multicovered space is a natural place where Theory of
Selection Principles develops naturally and deeply. A typical
selection principle asserts that for each sequence (u_{n})_{n Î w} Î m^{w} of covers of a multicovered space (X,m) it is
possible to select a cover v = {B_{n} : n Î w} of X by
u_{n}bounded subsets B_{n} Ì X so that for each point x Î X
the index set {n Î w: x Î B_{n}} is "large" in a suitable
sense. If "large" means "nonempty" (resp. "coinfinite") then
we obtain the classical Menger (resp. Hurewicz) property. The
(nontrivial and highly fruitful) interplay between Selection
Principles and recently created Theory of Semifilters will be
discussed as well.
 NIKOLAY BRODSKIY, University of Tennessee, Knoxville, TN
Compression of uniform embeddings into Hilbert space
[PDF] 
The notion of uniform embedding of metric spaces plays an important
role in study of large scale properties of finitely generated groups.
A map f: X® Y of metric spaces (X,d_{X}) and (Y,d_{Y}) is
called a uniform embedding if there are two real functions
r_{} and r_{+} with lim_{r®¥} r_{} (r) = +¥ such that r_{} (d_{X}(x,z)) £ d_{Y} (f(x),f(z)) £ r_{+}(d_{X}(x,z)) for all x,z Î X. For example, a biLipschitz map is a
uniform embedding with linear functions r_{} and r_{+}. If one
tries to embed a given space X uniformly into Hilbert space, how
close to biLipschitz could the embedding be? We answer this question
for finite dimensional CAT(0) cube complexes and for hyperbolic groups
with word metric.
 MAXIM BURKE, University of Prince Edward Island, Charlottetown, PE
C1A 4P3
Large entire crosssections of second category sets in
R^{n+1}
[PDF] 
By the KuratowskiUlam theorem, if A Í R^{n+1} = R^{n} ×R is a Borel set which has second
category intersection with every ball (i.e., is "everywhere
second category"), then there is a y Î R such that the
section AÇ(R^{n} ×{y}) is everywhere second
category in R^{n} ×{y}. If A is not Borel, then
there may not exist a large crosssection through A, even if the
section does not have to be flat. For example, a variation on a
result of T. Bartoszynski and L. Halbeisen shows that there is an
everywhere second category set A Í R^{n+1} such that
for any polynomial p in n variables, AÇgraph(p) is finite.
It is a classical result that under the Continuum Hypothesis, there is
an everywhere second category set L in R^{n+1} which has
only countably many points in any first category set. In particular,
LÇgraph(f) is countable for any continuous function f :R^{n} ® R. We prove that it is relatively
consistent with ZFC that for any everywhere second category set A in
R^{n+1}, there is a function f : R^{n} ®R which is the restriction to R^{n} of an entire
function on C^{n} and is such that, relative to graph(f),
the set AÇgraph(f) is everywhere second category. Moreover,
given a nonnegative integer k, a function g : R^{n}® R of class C^{k} and a positive continuous function
e: R^{n} ® R, we may choose f so
that for all multiindices a of order at most k and for all
x Î R^{n}, D^{a} f(x)D^{a} g(x) < e(x).
The method builds on fundamental work of K. Ciesielski and S. Shelah
which provides, for everywhere second category sets in 2^{w}×2^{w}, large sections which are the graphs of homeomorphism
of 2^{w}.
 ROBERT CAUTY, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05
Approximation by chain mappings and fixed points
[PDF] 
We use approximations of multivalued maps by chain mappings into
singular chain complexes to prove fixed point theorems.
 DOUGLAS CHILDERS, University of Alabama at Birmingham
Sequences of rotation numbers determine degeneracy of a
lamination on the closed unit disk
[PDF] 
Let S^{1} = R/Z denote the complex unit circle and
define s: S^{1} ® S^{1} by s(t) = 2t mod 1.
Thurston describes a collection of sinvariant laminations on
the complex unit disk [`(D)], which gives a
combinatoric parametrization of the Mandelbrot set. Each one of these
laminations defines an equivalence relation ~ on S^{1} such that
(s, ~ ) induces a map F : S^{1}/ ~ ® S^{1}/ ~ .
Often, there exists a quadratic polynomial P with Julia set J such
that P_{J} is semiconjugate to F. However there are obstructions
to this being true in general. One of these obstructions is that
S^{1}/ ~ could reduce to a point. In this case we call the
lamination degenerate.
Bullett and Sentenac introduced the notion of a closed set having a
sequence of rotation numbers for s. This notion is related to
"ouady tuning". We use this concept to give a necessary and
sufficient condition for when the lamination is degenerate.
 DALE DANIEL, Lamar University
Applications of selections to the HahnMazurkiewicz Problem
[PDF] 
The HahnMazurkiewicz Problem asks for conditions under which a
Hausdorff space is the continuous image of a generalized arc. The
first characterizations of continuous images of nonmetric arcs were
given by Bula and Turzanski and by Nikiel. Additional results include
those of Mardesi\'c, Treybig, and many others.
In a related study, we herein consider applications of selections
(carriers) to the study of images of ordered compacta. In particular,
let X be a compact ordered space, Y a Hausdorff space, and let
F(Y) denote the family of all nonempty closed subsets of Y with
the Vietoris topology. Assuming G : X® F(Y) is
continuous, we consider conditions under which G can be "lifted"
to a continuous map of X onto Y.
This work relies heavily on work of R. S. Countryman as well as the
theory of selections and that of continuous images of ordered
compacta.
 DIKRAN DIKRANJAN, Udine University
Compact groups without endomorphisms with infiniteentropy
[PDF] 
Let E denote the class of compact groups that admit no continuous
endomorphism with infinite topological entropy.
(1) There exist connected infinitedimensional groups in E,
but all abelian groups in E are finitedimensional.
(2) An abelian compact connected K group belongs to E iff
K is finitedimensional.
The nonconnected groups in E are hard to describe even in the
abelian case. In particular, we do not know the answer to the
following question: does there exist an abelian totally disconnected
group in E that has endomorphisms with positive entropy?
 XABIER DOMINGUEZ, Universidad de A Coruña, Campus de Elviña s/n, 15071
A Coruña, Spain
Asterisk topologies on the direct sum of topological Abelian
groups
[PDF] 
Let (G_{i})_{i Î I} be a family of topological Abelian groups and let
Å_{i Î I} G_{i} denote their algebraic direct sum, that is,
the subgroup of the product Õ_{i Î I} G_{i} formed by those
families x = (x(i))_{i Î I} for which x(i)=0 for all but finitely
many i Î I. If we let H^{Ù} denote the character group of
the Abelian group H, there are canonical algebraic isomorphisms

æ è


Õ
i Î I

G_{i} 
ö ø

Ù

» 
Å
i Î I

G_{i}^{Ù} 
æ è


Å
i Î I

G_{i} 
ö ø

Ù

» 
Õ
i Î I

G_{i}^{Ù}. 

Two significant group topologies arise on Å_{i Î I} G_{i} in
a very natural way: the box topology T_{b}, given by
"rectangular" neighborhoods of zero, and the coproduct topology
T_{f}, which is the finest group topology on
Å_{i Î I} G_{i} which makes all canonical inclusions
continuous.
Nevertheless, in general none of them turns the abovementioned
isomorphisms into topological ones, when considering Tychonoff
topology on the products, and compactopen topology on all dual
groups. The analysis of such duality properties leads to the
definition of an intermediate topology T_{*}, the socalled
asterisk topology, firstly introduced by Kaplan in 1948.
Actually the lack of a natural generalization of Minkowski functional
to groups gives rise to a number of variants of Kaplan's original
definition. We shall survey what is known about the conditions under
which such topologies are in fact the same, their behaviour with
respect to duality, reflexivity and local quasiconvexity, and their
relation with each other and with the box and coproduct topologies.
 JERZY DYDAK, University of Tennessee
Extensions of maps to the projective plane
[PDF] 
It is proved that for a 3dimensional compact metrizable space X
the infinite real projective space RP^{¥} is an absolute
extensor of X if and only if the real projective plane RP^{2} is an
absolute extensor of X.
 VITALI FEDORCHUK, Moscow State University, Moscow, Russia
Some new applications of resolutions
[PDF] 
The method of resolutions was introduced in [1] (see also [2]). This
method allows us to construct new spaces using given collections of
spaces. Many examples of applications of this method are given in
[3]. By applying iterated resolutions and fully closed mappings one
can obtain more sophisticated examples [4]. Here we present several
new applications of resolutions.
Theorem 1
For any prime p there exists a 2dimensional homogeneous separable
first countable compact space T_{p} such that dim(T_{p}×T_{q}) = 3 for p ¹ q.
Question 1
Are there homogeneous metrizable compacta X and Y such that dim(X×Y) < dimX + dimY?
Recent results by J. L. Bryant [5] imply that if X and Y are
homogeneous metrizable ANRcompacta, then
Question 2
Does the equality (1) hold if X is a homogeneous ANRcompactum and
Y is an arbitrary (homogeneous) metrizable compactum?
Remark 1
As for Question 1, we cannot omit homogeneity of Y, since
Pontryagin's surface P_{2} is homogeneous.
Another two results are joint with A. V. Ivanov and J. van Mill.
Theorem 2 [CH; [6]]
For every n Î N, there exists a family of separable
compacta X_{i}, i Î N, such that for every nonempty
finite subset M of N and every nonempty closed subset
F of P_{i Î M} X_{i} we have dimF = k(F)n, where k(F) is
integer such that k(F) ³ 1 for infinite F. Moreover, F=2^{c}
for infinite closed F.
Theorem 3 [CH; [6]]
There exists an infinite separable compactum X such that for any
positive integer m, if F is an infinite closed subset of X^{m},
then F=2^{c} and F is strongly infinitedimensional.
Question 3
Does there exist in ZFC an ndimensional compactum Y_{n}, n ³ 2,
such that for every m ³ 2, every nonempty closed subset F of
Y^{m}_{n} has dimension kn, where k is some integer between 0
and m?
Question 4
Does there exist in ZFC an infinitedimensional compactum Z such
that for every nonempty closed subset F of Z^{2} we have either
dimF = 0 or F is infinitedimensional?
References
 [1]

V. V. Fedorchuk,
Bicompacta with noncoinciding dimensionalities.
Soviet Math. Dokl. 9(1968), 11481150.
 [2]

V. V. Fedorchuk and K. P. Hart,
d23 Special Constructions.
In: Encyclopedia of General Topology (K. P. Hart, J. Nagata and
J. E. Vaughan, eds.), Elsevier Science Ltd., 2004, 229232.
 [3]

S. Watson,
The construction of topological spaces: planks and
resolutions.
In: Recent Progress in General Topology (M. Husek and J. van Mill,
eds.), NorthHolland Publishing Co., Amsterdam, 1992, 673757.
 [4]

V. V. Fedorchuk,
Fully closed mappings and their applications.
(Russian) Fundament. i Prikl. Matem. (4) 9(2003), 105235;
J. Math. Sci. (New York), to appear.
 [5]

J. L. Bryant,
Reflections on the BingBorsuk conjecture.
Preprint, 2003, 14.
 [6]

V. V. Fedorchuk, A. V. Ivanov and J. van Mill,
Intermediate dimensions of products.
Topology Appl., submitted.
 GARY GRUENHAGE, Auburn University, Auburn, AL, USA
The Baire property of function spaces with the compactopen
topology
[PDF] 
We discuss some recent results on the topic of the title.
 VALENTIN GUTEV, University of KwaZuluNatal
Completeness, sections and selections
[PDF] 
As a rule, most of the classical Michaeltype selection theorems for
the existence of singlevalued continuous selections are analogues
and, in certain respects, generalizations of ordinary extension
theorems. In contrast to this, most of the selection theorems for the
existence of semicontinuous setvalued selections seem to have no
proper analogues in the extension theory. In this talk, we will
discuss the role of the "selection" condition in such theorem, and
how it is related to the metrizability of the range of the
corresponding setvalued mappings.
 KAZUHIRO KAWAMURA, Institute of Mathematics, University of Tsukuba
Continuous (approximate) roots of continuous functions on
compacta
[PDF] 
For a compact Hausdorff space X, C(X) denotes the ring of all
continuous complexvalued functions on X. The ring C(X) is said
to be algebraically closed if each monic polynomial with
C(X)coefficients has a root in C(X). Starting with a classical
theorem due to Countryman, Jr., we discuss a problem on topological
characterizations of X with C(X) being algebraically closed. Also
the existence of "approximate roots" and related topics will be
discussed.
The present talk is based on joint works with A. Chigogidze,
A. Karasev, T. Miura and V. Valov.
 KRYSTYNA KUPERBERG, Auburn University, Auburn, AL 36849, USA
Wild and 2wild trajectories
[PDF] 
A trajectory of a flow on a 3manifold is wild if the closure of at
least one of the semitrajectories is a wild arc. A trajectory is
2wild if the closure of each semitrajectory is a wild arc.
We describe a method of embedding wild trajectories in flows on
3manifolds. This method yields interesting examples of dynamical
systems. In particular, every boundaryless 3manifold admits a flow
with a discrete set of fixed points and such that the closure of every
nontrivial trajectory is 2wild, which answers a question posed at the
2004 Spring Topology and Dynamics Conference.
 JOHN C. MAYER, University of Alabama at Birmingham
Thurston laminations of the unit disk, equivalence relations,
and polynomial Julia sets
[PDF] 
In a widely circulated preprint (1984) William Thurston introduced the
notion of a (geodesic) lamination of the unit disk. Laminations are
combinatorial/geometric/topological objects used to study Julia sets
of polynomials in analytic complex dynamics. A lamination of the unit
disk is a closed collection of chords of the disk that do not cross
each other (they may touch at endpoints). Consider the power map
f(z)=z^{d}, d > 1, on the unit circle; extend f linearly to the
lamination (the chords). A chord is critical if its endpoints map to
one point. A lamination is invariant if the collection maps to itself
forward and backward, with dmany disjoint preimages of each chord
backward, and f extends linearly to a positivelyoriented confluent
map of the disk to itself. The plan is that
(1) a lamination is determined by `pulling back' a set of
critical chords,
(2) the lamination naturally induces an equivalence relation
on the unit circle,
(3) the quotient space of the circle under this equivalence
relation is a topological Julia set, and
(4) the topological Julia set is dynamically (and
topologically) equivalent to an analytic Julia set for some degree
d polynomial.
But there are obstructions to the fulfillment of the plan. Thurston
completed most of the plan for d=2, but left some questions
unanswered. Moreover, fundamental questions remain unanswered for
d > 2, but recent progress has been made. In particular, one
obstruction is that the lamination determined by a collection of
critical chords may naturally induce a degenerate equivalence
relation, collapsing the circle to a point in the quotient. In this
talk, we show how the obstruction arises in degree d=2, and give
some insight into degree d=3 and greater. In a subsequent talk at
this meeting, D. Childers provides a complete solution to when
degeneracy occurs, for degree d=2, in terms of the dynamics of the
critical chord, answering an implicit question of Thurston.
This talk is mostly joint work with members of the UAB Laminations
Seminar: A. Blokh, L. Oversteegen, D. Childers, G. Brouwer, C. Curry,
and P. Eslami.
 CHRIS MOURON, Rhodes College
Periodic points of functions on simple triodlike continua
[PDF] 
A continuum X is simple triodlike if for every e > 0 there
exists a continuous function g_{e} : X® T
such that T is a simple triod and for every t Î T, diam((g_{e}(t))^{1} ) < e. I will discuss the
techniques used in showing when a map f : X® X
has a periodic point where X is a simple triodlike continuum.
 LEX OVERSTEEGEN, UAB, Birmingham, AL 35294, USA
On the structure of pseudo convex sets in the sphere
[PDF] 
A connected open subset U of the sphere is called pseudo convex if
for all points z in U there exist at most two closest points in
the boundary of U. Answering a question by David Herron and David
Minda, we show that such a set has at most two boundary components.
We also provide a detailed analysis of sets with this property.
 BRIAN RAINES, Baylor University, Waco, TX 767987328
Inverse limit spaces arising from problems in economics
[PDF] 
We discuss economic models for which the space of predicted future
states is an inverse limit space.
 JURIS STEPRANS, York University
Products of Sequential CLPcompact spaces
[PDF] 
The class of spaces which have the property that every cover by clopen
sets has a finite subcover was introduced by A. Sostaks. These spaces
are now known as CLPcompact spaces and it has emerged that much of
the interesting behaviour of this class derives from the possibility
that the product of two topological spaces contains clopen sets which
do not belong to the algebra generated by the product of the algebras
of clopen sets in each factor. Hence the productive nature of
CLPcompactness poses certain problems not occurring in the classical
case. Indeed, the problem of finding weak hypotheses under which the
product of CLPcompact spaces is CLPcompact should still be
considered to be open even though some progress has been recorded. It
will be shown that the product of finitely many sequential,
CLPcompact spaces is CLPcompact.
 PAUL SZEPTYCKI, York University, Toronto, ON M3J 1P3
Transversals of almost disjoint families
[PDF] 
For a family of sets A, a set X and a cardinal k (usually £ w), X is said to be a ktransversal of A if X Í ÈA and 0 ¹ aÇX < k for each a Î A. If k=2
we will say that X is a transversal of A. X is said to be a
Bernstein set for A if Æ ¹ aÇX ¹ a for each
a Î A. When an almost disjoint family admits a ktransversal or
a Bernstein set was first studied in [1] motivated mainly by
applications in topology.
We consider here a weaker property:
Definition
Given a family of sets A, A is said to admit a
stransversal if A can be written as A = È{A_{n} :n Î w} such that each A_{n} admits a transversal.
The restriction that an almost disjoint family admits a transversal is
quite strong and not of much interest. However, quite a wide class of
almost disjoint families admit stransversals. We consider the
question when an almost disjoint family admits a stransversal
and present some examples and applications.
References
 [1]

P. Erdös and A. Hajnal,
On a property of families of sets.
Acta Math. Acad. Sci. Hungar. 12(1961), 87124.

