We show, in particular, that a multivalued map $f$ from a closed subspace $X$
of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ has a point of period exactly
$M$ if and only if its continuous extension $\tilde f: \beta X\to {\rm
exp}_k(\beta \mathbb R^n)$ has such a point. The result also holds if one
repace $\mathbb R^n$ by a locally compact Lindel\"of space of finite dimension.
We also show that if $f$ is a colorable map froma normal space $X$ to the space
${\mathcal K}(X)$ of all compact subsets of $X$ then its extension $\tilde
f:\beta X\to {\mathcal K}(\beta X)$ is fixed-point free.

We introduce a method for analyzing high-dimensional data. Our approach is
inspired by Morse theory and uses the nudged elastic band method from
computational chemistry. As output, we produce an increasing sequence of cell
complexes modeling the dense regions of the data. We test the method on several
data sets and obtain small cell complexes revealing informative topological
structure.

Extending and unifying concepts extensively used in the literature, we
introduce the notion of approximable interpolation sets for algebras of
functions on locally compact groups, especially for weakly almost periodic
functions and for uniformly continuous functions. We characterize approximable
interpolation sets both in combinatorial terms and in terms of the
$\mathscr{LUC}$- and $\mathscr{WAP}$-compactifications and analyze some of
their properties.

Hereditary coreflective subcategories of an epireflective subcategory A of
Top such that I_2\notin A (here I_2 is the 2-point indiscrete space) were
studied in [C]. It was shown that a coreflective subcategory B of A is
hereditary (closed under the formation of subspaces) if and only if it is
closed under the formation of prime factors. The main problem studied in this
paper is the question whether this claim remains true if we study the (more
general) subcategories of A which are closed under topological sums and
quotients in A instead of the coreflective subcategories of A.

Using the property of being completely Baire, countable dense homogeneity and
the perfect set property we will be able, under Martin's Axiom for countable
posets, to distinguish non-principal ultrafilters on $\omega$ up to
homeomorphism. Here, we identify ultrafilters with subpaces of $2^\omega$ in
the obvious way. Using the same methods, still under Martin's Axiom for
countable posets, we will construct a non-principal ultrafilter $\UU\subseteq
2^\omega$ such that $\UU^\omega$ is countable dense homogeneous. This
consistently answers a question of Hru\v{s}\'ak and Zamora Avil\'es.

A topological space $Y$ is said to have (AEEP) if the following condition is
fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to
Y$ are two measurable functions, then the set $\Delta(f,g) = \{x \in X:\ f(x) =
g(x)\}$ is a member of $\mathfrak{M}$. It is shown that a metrizable space $Y$
has (AEEP) iff the cardinality of $Y$ is no greater than $2^{\aleph_0}$.

It is shown that if for a complete metric space $(X,d)$ there is a constant
$\epsilon > 0$ such that the intersection $\bigcap_{j=1}^n B_d(x_j,r_j)$ of
open balls is nonempty for every finite system $x_1,...,x_n \in X$ of centers
and a corresponding system of radii $r_1,...,r_n > 0$ such that $d(x_j,x_k)
\leqsl \epsilon$ and $d(x_j,x_k) < r_j + r_k$ ($j,k = 1,...,n$), then $X$ is an
ANR; and if in the above one may put $\epsilon = \infty$, the space $X$ is an
AR. A certain criterion for an incomplete metric space to be an A(N)R is
presented.

In this paper we introduce a new class of metric actions on separable (not
necessarily connected) metric spaces called "Cauchy-indivisible" actions. This
new class coincides with that of proper actions on locally compact metric
spaces and, as examples show, it may be different in general. The concept of
"Cauchy-indivisibility" follows a more general research direction proposal in
which we investigate the impact of basic notions in substantial results, like
the impact of local compactness and connectivity in the theory of proper
transformation groups.

In this paper we give some coupled ?fixed point results for mappings
satisfying different contractive conditions on complete partial metric spaces.

We prove that Michael's paraconvex-valued selection theorem for paracompact
spaces remains true for C'(E)-valued mappings defined on collectionwise normal
spaces. Some possible generalisations are also given.

Let $X$ be a Borel subset of the Cantor set \textbf{C} of additive or
multiplicative class ${\alpha},$ and $f: X \to Y$ be a continuous function with
compact preimages of points onto $Y \subset \textbf{C}.$ If the image $f(U)$ of
every clopen set $U$ is the intersection of an open and a closed set, then $Y$
is a Borel set of the same class. This result generalizes similar results for
open and closed functions.

A topological group $G$ is called an $M_\omega$-group if it admits a
countable cover $\K$ by closed metrizable subspaces of $G$ such that a subset
$U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every
$K\in\K$. It is shown that any two non-metrizable uncountable separable
zero-dimenisional $M_\omega$-groups are homeomorphic.

Given a function f: N --> (omega+1)-{0}, we say that a faithfully indexed
sequence {a_n: n in N} of elements of a topological group G is: (i) f-Cauchy
productive (f-productive) provided that the sequence {prod_{n=0}^m a_n^{z(n)}:
m in N} is left Cauchy (converges to some element of G, respectively) for each
function z: N --> Z such that |z(n)| <= f(n) for every n in N; (ii)
unconditionally f-Cauchy productive (unconditionally f-productive) provided
that the sequence {a_{s(n)}: n in N\} is (f\circ s)-Cauchy productive
(respectively, (f\circ s)-productive) for every bijection s: N --

The author proposes a method for investigating actions of finite groups on
aspherical spaces. Complete homotopy classification of free actions of finite
groups on aspherical spaces is obtained. Also there are some results about
non-free actions. For example a relation between the cohomology of finite
groups and the lattice structure of its subgroups is obtained by the proposed
method. This relation is formulated in terms of spectral sequences.

In this paper, we describe the relationship between the quasi-component q(G)
of a (perfectly) minimal pseudocompact abelian group G and the component
(\widetilde G)_0 of its completion. Specifically, we characterize the pairs
(C,A) of compact connected abelian groups C and subgroups A such that A=q(G)
and C=(\widetilde G)_0. As a consequence, we show that for every positive
integer n or n=\omega, there exist plenty of abelian pseudocompact perfectly
minimal n-dimensional groups G such that the quasi-component of G is not dense
in the connected component of the completion of G.

We establish coupled fixed point theorems for contraction involving rational
expressions in partially ordered metric spaces.

The goal of this article is two folds. First, to introduce some problems
about structural identities of natural materials, in particular, collagen
networks; materials found everywhere around us. Second, to introduce some
beautiful known riddles in intuitive topology, and commenting about existence
of connections between field of topology and our first goal.

Properties of circulant graphs have been studied by many authors, but just a
few results concerning their genus characterization were presented up to now.
We can quote the classification of all circulant planar graphs given by C.
Heuberger in 2003. We present here a complete classification of circulant
graphs of genus one, derive a general lower bound for the genus of a circulant
graph and construct a family of circulant graphs which reach this bound.

In this paper we investigate the functors of OH of positively homogenous
functionals and OS of semiadditive functionals. We show that OH(X) is AR if and
only if X is openly generated, and OS(X) is AR if and only if X is an openly
generated compactum of weight less than $\omega_1$. Also, we investigate the
multiplication maps of monads generated by the abovementioned functors and
consider when these mappings are soft.

Iwasa investigated the preservation of various covering properties of
opological spaces under Cohen forcing. By improving the argument in Iwasa's
paper, we prove that the Rothberger property, the Menger property and selective
screenability are also preserved under Cohen forcing and forcing with the
measure algebra.

It has been claimed by Halmos in [Comment on the real line, Bull. Amer. Math.
Soc., 50 (1944), 877-878] that if G is a Hausdorff locally compact topological
abelian group and if the character group of G is torsion free then G is
divisible. We prove that such claim is false, by presenting a counterexample.
We also present a stronger counterexample, showing that even if one assumes
that the character group of G is both torsion free and divisible, it does not
follow that G is divisible.

We prove that for a complete quasivariety $K$ of topological $E$-algebras of
countable discrete signature $E$ and each submetrizable $ANR(k_\omega)$-space
$X$ its free topological $E$-algebra $F_K(X)$ in the class $K$ is a
submetrizable $ANR(k_\omega)$-space.

Many classes of maps are characterized as (possibly multi-valued) maps
preserving particular types of compact filters.

We use geometric techniques to explicitly find the topological structure of
the space of SO(3)-representations of the fundamental group of a closed surface
of genus 2 quotient by the conjugation action by SO(3). There are two
components of the space. We will describe the topology of both components and
describe the corresponding SU(2)-character spaces. There is the sixteen to one
branch-covering for each component, and the branch locus is a union of
2-spheres and 2-tori. Along the way, we also describe the topology of both
spaces.

Conditions on a topological space $X$ under which the space $C(X,\mathbb{R})$
of continuous real-valued maps with the Isbell topology $\kappa $ is a
topological group (topological vector space) are investigated. It is proved
that the addition is jointly continuous at the zero function in
$C_{\kappa}(X,\mathbb{R})$ if and only if $X$ is infraconsonant. This property
is (formally) weaker than consonance, which implies that the Isbell and the
compact-open topologies coincide.

We investigate preservation of the Lindel\"of property of topological spaces
under forcing extensions. We give sufficient conditions for a forcing notion to
preserve several strengthenings of the Lindel\"of property, such as
indestructible Lindel\"of property, the Rothberger property and being a
Lindel\"of P-space.

We present a series of examples of precompact, noncompact, reflexive
topological Abelian groups. Some of them are pseudocompact or even countably
compact, but we show that there exist precompact non-pseudocompact reflexive
groups as well. It is also proved that every pseudocompact Abelian group is a
quotient of a reflexive pseudocompact group with respect to a closed reflexive
pseudocompact subgroup.

Let S be a surface of negative Euler characteristic together with a pants
decomposition P. Kra's plumbing construction endows S with a projective
structure as follows. Replace each pair of pants by a triply punctured sphere
and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of
the punctures. The gluing across the $i^{th}$ pants curve is defined by a
complex parameter t_i in C. The associated holonomy representation \rho:
\pi_1(S) \to PSL(2,C) gives a projective structure on S which depends
holomorphically on the t_i.

Revisiting and completing a work due to A. I. Ba\v{s}kirov, we construct
compact sequential spaces of any sequential order up to and including
$\omega_1$ as quotient spaces of $\beta\omega$ under CH.

The recent extensions of domain theory have proved particularly efficient to
study lattice-valued maxitive measures, when the target lattice is continuous.
Maxitive measures are defined analogously to classical measures with the
supremum operation in place of the addition. Building further on the links
between domain theory and idempotent analysis highlighted by Lawson (2004), we
introduce the concept of domain-valued maxitive maps, which we define as a
``point-free'' version of maxitive measures.

In addition to 29 announcements in related areas, this issue contains several
contributions to "core" SPM: Measurable cardinals and the cardinality of
Lindelof spaces; Topological games and covering dimension; Menger's and
Hurewicz's Problems: Solutions from "The Book" and refinements; Point-cofinite
covers in the Laver model; Projective versions of selection principles

A Hausdorff topological group $(G,\tau)$ is called locally minimal if there
exists a neighborhood $U$ of 0 in $\tau$ such that $U$ fails to be a
neighborhood of zero in any Hausdorff group topology on $G$ which is strictly
coarser than $\tau.$ Examples of locally minimal groups are all subgroups of
Banach-Lie groups, all locally compact groups and all minimal groups.

For an uncountable cardinal \tau and a subset S of an abelian group G, the
following conditions are equivalent: (i) |{ns:s\in S}|\ge \tau for all integers
n\ge 1; (ii) there exists a group homomorphism \pi:G\to T^{2^\tau} such that
\pi(S) is dense in T^{2^\tau}. Moreover, if |G|\le 2^{2^\tau}, then the
following item can be added to this list: (iii) there exists an isomorphism
\pi:G\to G' between G and a subgroup G' of T^{2^\tau} such that \pi(S) is dense
in T^{2^\tau}.

Suppose G is a topological group containing a (closed) topological copy of
the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal
k-space) then every closed metrizable subset in $G$ is locally compact.
Applying this result to topological groups whose underlying topological space
can be written as a direct limit of a sequence of closed metrizable subsets, we
get that every such a group either is metrizable or is homeomorphic to the
product of a $k_\omega$-space and a discrete space.

Positive Quaternion Kaehler Manifolds are Riemannian manifolds with holonomy
contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they
are symmetric spaces. We offer a new approach to this field of study via
Rational Homotopy Theory, thereby proving the formality of Positive Quaternion
Kaehler Manifolds. This result is established by means of an in-depth
investigation on how formality behaves under spherical fibrations.

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 zero-dimensional space $X$ there exists a zero-dimensional
space $Y$ such that $X\times Y$ is h-homogeneous. Also, we simultaneously
generalize results of Motorov and Terada by showing that if $X$ is a
zero-dimensional space such that the isolated points are dense then $X^\kappa$
is h-homogeneous for every infinite cardinal $\kappa$.

In this brief note we provide a simple approach to show that the Alaoglu
theorem and the Tychonoff theorem for compact Hausdorff spaces are equivalent.

In this note we study the dynamics of the natural evaluation action of the
group of isometries $G$ of a locally compact metric space $(X,d)$ with one end.
Using the notion of pseudo-components introduced by S. Gao and A. S. Kechris we
show that $X$ has only finitely many pseudo-components of which exactly one is
not compact and $G$ acts properly on. The complement of the non-compact
component is a compact subset of $X$ and $G$ may fail to act properly on it.

We prove that the Khovanov homology of alternating knots and 2-component
links is equal (as a singly graded group) to the singular homology of a certain
space of trace- free, binary dihedral representations of the link group. More
generally, it was suggested by Kronheimer and Mrowka that the Khovanov homology
of any knot might be related via gauge theory to the space of all trace-free
SU(2) representations. Our result suggests that when the knot is alternating,
Khovanov homology only sees the trace-free representations which are binary
dihedral.

Hurewicz' characterized the dimension of separable metrizable spaces by means
of finite-to-one maps. We investigate whether this characterization also holds
in the class of compact F-spaces of weight c. Our main result is that, assuming
the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the
continuous image of a zero-dimensional compact Hausdorff space by an at most
2n-to-1 map.

We give a construction under $CH$ of a non-metrizable compact Hausdorff space
$K$ such that any uncountable semi-biorthogonal sequence in $C(K)$ must be of a
very specific kind. The space $K$ has many nice properties, such as being
hereditarily separable, hereditarily Lindel\"of and a 2-to-1 continuous
preimage of a metric space, and all Radon measures on $K$ are separable.
However $K$ is not a Rosenthal compactum.

We investigate contrasting behaviours emerging when studying foliations on
non-metrisable manifolds. It is shown that Kneser's pathology of a manifold
foliated by a single leaf cannot occur with foliations of dimension-one. On the
other hand, there are open surfaces admitting no foliations.

Manifolds have uses throughout and beyond Mathematics and it is not
surprising that topologists have expended a huge effort in trying to understand
them. In this article we are particularly interested in the question: `when is
a manifold metrisable?' We describe many conditions equivalent to
metrisability.

We investigate the mapping class group of an orientable $\omega$-bounded
surface. Such a surface splits, by Nyikos's Bagpipe Theorem, into a union of a
bag (a compact surface with boundary) and finitely many long pipes. The
subgroup consisting of classes of homeomorphisms fixing the boundary of the bag
is a normal subgroup and is a homomorphic image of the product of mapping class
groups of the bag and the pipes.

The recent literature offers examples, specific and hand-crafted, of
Tychonoff spaces (in ZFC) which respond negatively to these questions, due
respectively to Ceder and Pearson (1967) and to Comfort and Garc\'ia-Ferreira
(2001): (1) Is every $\omega$-resolvable space maximally resolvable? (2) Is
every maximally resolvable space extraresolvable? Now using the method of
${\mathcal{KID}}$ expansion, the authors show that {\it every} suitably
restricted Tychonoff topological space $(X,\sT)$ admits a larger Tychonoff
topology (that is, an "expansion") witnessing such failure.

Bornological universes were introduced some time ago by Hu and obtained
renewed interest in recent articles on convergence in hyperspaces and function
spaces and optimization theory. One of Hu's results gives us a necessary and
sufficient condition for which a bornological universe is metrizable. In this
article we will extend this result and give a characterization of uniformizable
bornological universes.

A family $\bfam$ of continuous real-valued functions on a space $X$ is said
to be {\sl basic} if every $f \in C(X)$ can be represented $f = \sum_{i=1}^n
g_i \circ \phi_i$ for some $\phi_i \in \bfam$ and $g_i \in C(\R)$ ($i=1, ...,
n$). Define $\basic (X) = \min \{|\bfam| : \bfam$ is a basic family for $X\}$.
If $X$ is separable metrizable $X$ then either $X$ is locally compact and
finite dimensional, and $\basic (X) < \aleph_0$, or $\basic (X) =
\mathfrak{c}$.

We extend the theory of Euler integration from the class of constructible
functions to that of "tame" real-valued functions (definable with respect to an
o-minimal structure). The corresponding integral operator has some unusual
defects (it is not a linear operator); however, it has a compelling
Morse-theoretic interpretation. In addition, we show that it is an appropriate
setting in which to do numerical analysis of Euler integrals, with applications
to incomplete and uncertain data in sensor networks.

The natural quotient map q from the space of based loops in the Hawaiian
earring onto the fundamental group provides a new and naturally occuring
example of a quotient map such that q x q fails to be a quotient map.

This counterexample also contradicts a number of published claims, notably
pi1(X,p) can in fact fail to be a topological group.

In a recent paper by D. Shakhmatov and J. Sp\v{e}v\'ak [Group-valued
continuous functions with the topology of pointwise convergence, Topology and
its Applications (2009), doi:10.1016/j.topol.2009.06.022] the concept of a
${\rm TAP}$ group is introduced and it is shown in particular that the ${\rm
NSS}$ groups are ${\rm TAP}$. We prove that conversely, the Weil complete
metrizable ${\rm TAP}$ groups are ${\rm NSS}$. We define also the narrower
class of ${\rm STAP}$ groups, show that the ${\rm NSS}$ groups are if fact
${\rm STAP}$ and that the converse statement is true in metrizable case.

Generalizing a theorem of Ph. Dwinger, we describe the ordered set of all (up
to equivalence) zero-dimensional locally compact Hausdorff extensions of a
zero-dimensional Hausdorff space. Using this description, we find the necessary
and sufficient conditions which has to satisfy a map between two
zero-dimensional Hausdorff spaces in order to have some kind of extension over
two given Hausdorff zero-dimensional local compactifications of these spaces;
we regard the following kinds of extensions: continuous, open, quasi-open,
skeletal, perfect, injective, surjective.

Arhangel'skii proved that if a first countable Hausdorff space is Lindel\"of,
then its cardinality is at most $2^{\aleph_0}$. Such a clean upper bound for
Lindel\"of spaces in the larger class of spaces whose points are ${\sf
G}_{\delta}$ has been more elusive. In this paper we continue the agenda
started in F.D. Tall, On the cardinality of Lindel\"of spaces with points
$G_{\delta}$, Topology and its Applications 63 (1995), 21 - 38, of considering
the cardinality problem for spaces satisfying stronger versions of the
Lindel\"of property.

We show that Lelek's problem on the chainability of continua with span zero
is not a metric problem: from a non-metric counterexample one can construct a
metric one.

Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is
a closed subspace in $X_{n+1}$. We give an explicit description of the topology
and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the
category of uniform spaces.