1. CMB 2016 (vol 59 pp. 673)
 Bačák, Miroslav; Kovalev, Leonid V.

Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups
Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric
space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped
with the Hausdorff metric is called a finite subset space. In
this paper we are concerned with the existence of Lipschitz retractions
$r\colon X(n)\to X(n1)$ for $n\ge2.$ It is known that such retractions
do not exist if $X$ is the onedimensional sphere. On the other
hand L. Kovalev has recently established their existence in case $X$
is a Hilbert space and he also posed a question as to whether
or not such Lipschitz retractions exist for $X$ being a Hadamard
space. In the present paper we answer this question in the positive.
Keywords:finite subset space, gradient flow, Hadamard space, LieTrotterKato formula, Lipschitz retraction Categories:53C23, 47H20, 54E40, 58D07 

2. CMB 2016 (vol 59 pp. 693)
 Chen, ChungChuan

Recurrence of Cosine Operator Functions on Groups
In this note, we study the recurrence and topologically multiple
recurrence of a sequence of operators on Banach spaces.
In particular, we give a sufficient and necessary condition for
a cosine operator function,
induced by a sequence of operators on the Lebesgue space of a
locally compact group, to be topologically multiply recurrent.
Keywords:topologically multiple recurrence, recurrence, topological transitivity, hypercyclicity, cosine operator function Categories:47A16, 54B20, 43A15 

3. CMB 2015 (vol 58 pp. 334)
 Medini, Andrea

Countable Dense Homogeneity in Powers of Zerodimensional Definable Spaces
We show that, for a coanalytic subspace $X$ of $2^\omega$, the
countable dense homogeneity of $X^\omega$ is equivalent to $X$
being Polish. This strengthens a result of HruÅ¡Ã¡k and Zamora
AvilÃ©s. Then, inspired by results of HernÃ¡ndezGutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a nonPolish subspace $X$ of $2^\omega$ such that $X^\omega$
is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer
to a question of HruÅ¡Ã¡k and Zamora AvilÃ©s. 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 zerodimensional separable metrizable space $X$ is
included in a Polish subspace of $X$ then $X^\omega$ is countable
dense homogeneous.
Keywords:countable dense homogeneous, infinite power, coanalytic, Polish, $\lambda'$set Categories:54H05, 54G20, 54E52 

4. CMB 2014 (vol 58 pp. 7)
 Boulabiar, Karim

Characters on $C(X)$
The precise condition on a completely regular space $X$ for every character on
$C(X) $ to be an evaluation at some point in $X$ is that $X$ be
realcompact. Usually, this classical result is obtained relying heavily on
involved (and even nonconstructive) extension arguments. This note provides a
direct proof that is accessible to a large audience.
Keywords:characters, realcompact, evaluation, realvalued continuous functions Categories:54C30, 46E25 

5. CMB 2014 (vol 57 pp. 579)
 Larson, Paul; Tall, Franklin D.

On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces
We establish that if it is consistent that there is a
supercompact cardinal, then it is consistent that every locally
compact, hereditarily normal space which does not include a perfect
preimage of $\omega_1$ is hereditarily paracompact.
Keywords:locally compact, hereditarily normal, paracompact, Axiom R, PFA$^{++}$ Categories:54D35, 54D15, 54D20, 54D45, 03E65, 03E35 

6. CMB 2014 (vol 57 pp. 803)
 Gabriyelyan, S. S.

Free Locally Convex Spaces and the $k$space Property
Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$.
Keywords:free locally convex space, $k$space, countable tightness Categories:46A03, 54D50, 54A25 

7. CMB 2014 (vol 57 pp. 683)
 Aurichi, Leandro F.; Dias, Rodrigo R.

Topological Games and Alster Spaces
In this paper we study connections between topological games
such
as Rothberger, Menger and compactopen, and relate these games
to
properties involving covers by $G_\delta$ subsets. The results
include:
(1) If Two has a winning strategy in the Menger
game on a regular space $X$, then $X$ is an Alster space.
(2) If Two has a winning strategy in the Rothberger game on a
topological space $X$, then the $G_\delta$topology on $X$ is
LindelÃ¶f.
(3) The Menger game and the compactopen game are (consistently)
not
dual.
Keywords:topological games, selection principles, Alster spaces, Menger spaces, Rothberger spaces, Menger game, Rothberger game, compactopen game, $G_\delta$topology Categories:54D20, 54G99, 54A10 

8. CMB 2013 (vol 57 pp. 631)
 Sokić, Miodrag

Indicators, Chains, Antichains, Ramsey Property
We introduce two Ramsey classes of finite relational structures. The first
class contains finite structures of the form $(A,(I_{i})_{i=1}^{n},\leq
,(\preceq _{i})_{i=1}^{n})$ where $\leq $ is a total ordering on $A$ and $%
\preceq _{i}$ is a linear ordering on the set $\{a\in A:I_{i}(a)\}$. The
second class contains structures of the form $(A,\leq
,(I_{i})_{i=1}^{n},\preceq )$ where $(A,\leq )$ is a weak ordering and $%
\preceq $ is a linear ordering on $A$ such that $A$ is partitioned by $%
\{a\in A:I_{i}(a)\}$ into maximal chains in the partial ordering $\leq $ and
each $\{a\in A:I_{i}(a)\}$ is an interval with respect to $\preceq $.
Keywords:Ramsey property, linear orderings Categories:05C55, 03C15, 54H20 

9. CMB 2013 (vol 57 pp. 245)
 Brodskiy, N.; Dydak, J.; Lang, U.

AssouadNagata Dimension of Wreath Products of Groups
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated.
We show that the AssouadNagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$
depends on the growth of $G$ as follows:
\par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$,
otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.
Keywords:AssouadNagata dimension, asymptotic dimension, wreath product, growth of groups Categories:54F45, 55M10, 54C65 

10. CMB 2013 (vol 57 pp. 364)
 Li, Lei; Wang, YaShu

How Lipschitz Functions Characterize the Underlying Metric Spaces
Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that
both $X,Y$ are realcompact, or both $E,F$ are realcompact.
The zero set of a vectorvalued function $f$ is denoted by $z(f)$.
A linear bijection $T$ between local or generalized Lipschitz vectorvalued function spaces
is said to preserve zeroset containments or nonvanishing functions
if
\[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\]
or
\[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\]
respectively.
Every zeroset containment preserver, and every nonvanishing function preserver when
$\dim E =\dim F\lt +\infty$, is a weighted composition operator
$(Tf)(y)=J_y(f(\tau(y)))$.
We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.
Keywords:(generalized, locally, little) Lipschitz functions, zeroset containment preservers, biseparating maps Categories:46E40, 54D60, 46E15 

11. CMB 2013 (vol 57 pp. 335)
 Karassev, A.; Todorov, V.; Valov, V.

Alexandroff Manifolds and Homogeneous Continua
ny homogeneous,
metric $ANR$continuum is a $V^n_G$continuum provided $\dim_GX=n\geq
1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal
domain.
This implies that any homogeneous $n$dimensional metric $ANR$continuum is a $V^n$continuum in the sense of Alexandroff.
We also prove that any finitedimensional homogeneous metric continuum
$X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq
1$, cannot be separated by
a compactum $K$ with $\check{H}^{n1}(K;G)=0$ and $\dim_G K\leq
n1$. This provides a partial answer to a question of
KallipolitiPapasoglu
whether any twodimensional homogeneous Peano continuum cannot be separated by arcs.
Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$continuum Categories:54F45, 54F15 

12. CMB 2012 (vol 57 pp. 240)
 Bernardes, Nilson C.

Addendum to ``Limit Sets of Typical Homeomorphisms''
Given an integer $n \geq 3$,
a metrizable compact topological $n$manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 

13. CMB 2012 (vol 56 pp. 709)
 Bartošová, Dana

Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a wellknown fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using FraÃ¯ssÃ© theory and Ramsey theory. This work
generalizes some of the known results about countable structures.
Keywords:universal minimal flows, ultrafilter flows, Ramsey theory Categories:37B05, 03E02, 05D10, 22F50, 54H20 

14. CMB 2012 (vol 56 pp. 860)
 van Mill, Jan

On Countable Dense and $n$homogeneity
We prove that a connected, countable dense homogeneous space is
$n$homogeneous for every $n$, and strongly 2homogeneous provided it
is locally connected. We also present an example of a connected and
countable dense homogeneous space which is not strongly
2homogeneous. This answers Problem 136 of Watson in the Open Problems
in Topology Book in the negative.
Keywords:countable dense homogeneous, connected, $n$homogeneous, strongly $n$homogeneous, counterexample Categories:54H15, 54C10, 54F05 

15. CMB 2011 (vol 56 pp. 292)
 Dai, MeiFeng

Quasisymmetrically Minimal Moran Sets
M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor
sets of Hausdorff dimension $1$, where at the $k$th set one removes
from each interval $I$ a certain number $n_{k}$ of open subintervals
of length $c_{k}I$, leaving $(n_{k}+1)$ closed subintervals of
equal length. Quasisymmetrically Moran sets of Hausdorff dimension $1$
considered in the paper are more general than uniform Cantor sets in
that neither the open subintervals nor the closed subintervals are
required to be of equal length.
Keywords:quasisymmetric, Moran set, Hausdorff dimension Categories:28A80, 54C30 

16. CMB 2011 (vol 56 pp. 442)
 Zelenyuk, Yevhen

Closed Left Ideal Decompositions of $U(G)$
Let $G$ be an infinite discrete group and let $\beta G$ be the
StoneÄech compactification of $G$. We take the points of $Äta
G$ to be the ultrafilters on $G$, identifying the principal
ultrafilters with the points of $G$. The set $U(G)$ of uniform
ultrafilters on $G$ is a closed twosided ideal of $\beta G$. For
every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in
p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show
that if $G$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the
finest decomposition of $U(G)$ into closed left ideals of $\beta G$
such that the corresponding quotient space of $U(G)$ is Hausdorff.
Keywords:StoneÄech compactification, uniform ultrafilter, closed left ideal, decomposition Categories:22A15, 54H20, 22A30, 54D80 

17. CMB 2011 (vol 56 pp. 92)
 Jacob, Benoît

On Perturbations of Continuous Maps
We give sufficient conditions for the following problem: given a
topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and
a continuous map $f$ from $X$ to $Y$, is it possible, by applying to
$f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not
meet $Z$? We also give a relative variant: if $f(X')$ does not meet
$Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged
on $X'$. We also develop a variant for continuous sections of
fibrations and discuss some applications to matrix perturbation
theory.
Keywords:perturbation theory, general topology, applications to operator algebras / matrix perturbation theory Category:54F45 

18. CMB 2011 (vol 56 pp. 424)
 Thom, Andreas

Convergent Sequences in Discrete Groups
We prove that a finitely generated group contains a
sequence of nontrivial elements that converge to the identity in
every compact homomorphic image if and only if the group is not
virtually abelian. As a consequence of the methods used, we show that a finitely generated
group satisfies Chu duality if and only if it is virtually abelian.
Keywords:Chu duality, Bohr topology Category:54H11 

19. CMB 2011 (vol 56 pp. 203)
 Tall, Franklin D.

Productively LindelÃ¶f Spaces May All Be $D$
We give easy proofs that (a) the Continuum Hypothesis implies that if
the product of $X$ with every LindelÃ¶f space is LindelÃ¶f, then $X$ is
a $D$space, and (b) Borel's Conjecture implies every Rothberger space
is Hurewicz.
Keywords:productively LindelÃ¶f, $D$space, projectively $\sigma$compact, Menger, Hurewicz Categories:54D20, 54B10, 54D55, 54A20, 03F50 

20. CMB 2011 (vol 56 pp. 55)
 Bouziad, A.

Cliquishness and Quasicontinuity of TwoVariable Maps
We study the existence of continuity points for mappings
$f\colon X\times Y\to Z$ whose $x$sections $Y\ni y\to f(x,y)\in Z$ are
fragmentable and $y$sections $X\ni x\to f(x,y)\in Z$ are
quasicontinuous, where $X$ is a Baire space and $Z$
is a metric space. For the factor $Y$, we consider two
infinite ``pointpicking'' games $G_1(y)$ and $G_2(y)$ defined respectively
for each $y\in Y$ as follows: in the $n$th
inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then
Player II picks a point $y_n\in D_n$;
II wins if $y$ is in the closure of ${\{y_n:n\in\mathbb N\}}$, otherwise
I wins. It is shown that
(i) $f$ is
cliquish
if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $
f$ is quasicontinuous if
the $x$sections of $f$ are continuous and the set of $y\in Y$
such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially
a result of Debs and item (ii) indicates that
the problem of Talagrand on separately continuous maps has a positive answer for a wide
class of ``small'' compact spaces.
Keywords:cliquishness, fragmentability, joint continuity, pointpicking game, quasicontinuity, separate continuity, two variable maps Categories:54C05, 54C08, 54B10, 91A05 

21. CMB 2011 (vol 55 pp. 297)
 Glasner, Eli

The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact
Following a similar result of Uspenskij on the unitary group of a
separable Hilbert space, we show that, with respect to the lower (or
Roelcke) uniform structure, the Polish group $G=
\operatorname{Aut}(\mu)$ of automorphisms of an atomless standard
Borel probability space $(X,\mu)$ is precompact. We identify the
corresponding compactification as the space of Markov operators on
$L_2(\mu)$ and deduce that the algebra of right and left uniformly
continuous functions, the algebra of weakly almost periodic functions,
and the algebra of Hilbert functions on $G$, i.e., functions on
$G$ arising from unitary representations, all coincide. Again
following Uspenskij, we also conclude that $G$ is totally minimal.
Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions Categories:54H11, 22A05, 37B05, 54H20 

22. CMB 2011 (vol 55 pp. 225)
 Bernardes, Nilson C.

Limit Sets of Typical Homeomorphisms
Given an integer $n \geq 3$, a metrizable compact
topological $n$manifold $X$ with boundary, and a finite positive Borel
measure $\mu$ on $X$, we prove that for the typical homeomorphism
$f \colon X \to X$, it is true that for $\mu$almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
nonsensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 

23. CMB 2011 (vol 54 pp. 607)
 Camargo, Javier

Lightness of Induced Maps and Homeomorphisms
An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and
$2^{f}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive
integer $n$, we study when the lightness of the induced map $2^{f}$ or $C_n(f)$ implies that $f$ is a
homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.
Keywords:light maps, induced maps, continua, hyperspaces Categories:54B20, 54E40 

24. CMB 2011 (vol 54 pp. 302)
25. CMB 2011 (vol 54 pp. 244)
 Daniel, D. ; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.

Homogeneous Suslinian Continua
A continuum is said to be Suslinian if it does not
contain uncountably many
mutually exclusive nondegenerate subcontinua. Fitzpatrick and
Lelek have shown that a metric Suslinian continuum $X$ has the
property that the set of points at which $X$ is connected im
kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua
and obtain a number of corollaries. In particular, we prove that a homogeneous,
nondegenerate, Suslinian continuum is a simple closed curve and that each separable,
nondegenerate, homogenous, Suslinian continuum is metrizable.
Keywords:connected im kleinen, homogeneity, Suslinian, locally connected continuum Categories:54F15, 54C05, 54F05, 54F50 
