The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erd\H{o}s), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n{-}1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\setminus S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets, is shown to be closely related to Topological Vaught's Conjecture.

Extending the work of Larson and Todorcevic, we show there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$ is a coherent Souslin tree.

In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT$(0)$ spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.

A new decomposition, the \emph{mutually aposyndetic decomposition} of homogeneous continua into closed, homogeneous sets is introduced. This decomposition is respected by homeomorphisms and topologically unique. Its quotient is a mutually aposyndetic homogeneous continuum, and in all known examples, as well as in some general cases, the members of the decomposition are semi-indecomposable continua. As applications, we show that hereditarily decomposable homogeneous continua and path connected homogeneous continua are mutually aposyndetic. A class of new examples of homogeneous continua is defined. The mutually aposyndetic decomposition of each of these continua is non-trivial and different from Jones' aposyndetic decomposition.

If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\operatorname{Spec} H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.

Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight $\aleph_1$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following. \begin{compactenum}[\rm(a)] \item Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets. \item If $G$ is a countably infinite Abelian group with finitely many elements of order $2$ and $\beta G$ is the Stone--\v Cech compactification of $G$ as a discrete semigroup, then for every idempotent $p\in\beta G\setminus\{0\}$, the subset $\{p,-p\}\subset\beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and~$\{-p\}$. \end{compactenum}

The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}.

We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of $\R^{N-1}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in $\R^d$ in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris-Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative rings (with units). We call such an embedding a $\CR$-epic embedding and we say that $X$ is absolute $\CR$-epic if every embedding of $X$ is $\CR$-epic. We continue this investigation. Our most notable result shows that a Lindel\"of space $X$ is absolute $\CR$-epic if a countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta X$-neighbourhood of $X$. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindel\"of property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all \s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger condition. We get some results in the non-Lindel\"of case that are sufficient to show that the Dieudonn\'e plank and some closely related spaces are absolute $\CR$-epic.

We study Tychonoff spaces $X$ with the property that, for all topological embeddings $X\to Y $, the induced map $C(Y) \to C(X)$ is an epimorphism of rings. Such spaces are called \good. The simplest examples of \good spaces are $\sigma$-compact locally compact spaces and \Lin $P$-spaces. We show that \good first countable spaces must be locally compact. However, a ``bad'' class of \good spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not \good abound, and some are presented.

Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.

Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of (complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists $\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

Giordano, Putnam and Skau showed that the transformation group $C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C(X)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result.

If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha)$ be the space of all continuous real-valued functions defined on the ordinal interval $[0, \alpha]$. We characterize the symmetric sequence spaces which embed into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy $(E_\alpha)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, $E_\alpha$ embeds into $C(\omega^{\omega^\alpha})$, but does not embed into $C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.

The notion of shape fibration with the near lifting of near multivalued paths property is studied. The relation of these maps---which agree with shape fibrations having totally disconnected fibers---with Hurewicz fibrations with the unique path lifting property is completely settled. Some results concerning homotopy and shape groups are presented for shape fibrations with the near lifting of near multivalued paths property. It is shown that for this class of shape fibrations the existence of liftings of a fine multivalued map, is equivalent to an algebraic problem relative to the homotopy, shape or strong shape groups associated.

We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.