General Topology

We consider special subclasses of the class of Lindel\"of Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $L\Sigma(\leq\kappa)$ if it…

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. We denote this class by R. Allowing continuous images in the definition of class R, one…

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of ``simple'' retractions. Consequently, a compact space of weight $\loe\aleph_1$ is Valdivia compact iff it is the limit of an…

We investigate the class of compact spaces which are embeddable into a power of the real line $R^\kappa$ in such a way that c_0(\kappa) is dense in the image. We show that this is a proper subclass of the class of Valdivia, even when…

A compact space $K$ is {\em Valdivia compact} if it can be embedded in a Tikhonov cube $I^A$ in such a way that the intersection $K\cap\Sigma$ is dense in $K$, where $\Sigma$ is the sigma-product (= the set of points with countably many…

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a…

We study some topological properties of trees with the interval topology. In particular, we characterize trees which admit a 2-fibered compactification and we present two examples of trees whose one-point compactifications are Rosenthal…

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The…

Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all…

Let R denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. Class R is striclty larger than the class of Valdivia compact spaces. We show that…

Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\cup…

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of…

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is…

A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally disconnected, compact metric space without isolated points. A 1920 theorem of Sierpinski characterizes the rationals as the unique countable metric space without…

In this paper we define fuzzy sequential topology on a non empty set X which is a collection of fuzzy sequential sets satisfying certain conditions given in its definition and many pleasant properties of a countable number of fuzzy…

A space is `n-strong arc connected' (n-sac) if for any n points in the space there is an arc in the space visiting them in order. A space is omega-strong arc connected (omega-sac) if it is n-sac for all n. We study these properties in…

A space is `n-arc connected' (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the…

We show that if a topological or paratopological group $G$ contains a stationary subset of some regular uncountable cardinal, then $G$ contains a subspace which is not collectionwise normal. This statement implies that if a monotonically…

We prove the following result announced in Todorov and Valov: Any 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…

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of…