Square compactness and the filter extension property

We show that the consistency strength of κ being 2κ-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juhasz

The Tamano theorem in MAP

In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category TOP (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also given.peer-reviewe

A category of continuous maps

The study of General Topology is usually concerned with the category TOP of topological spaces as objects, and continuous maps as morphisms. The concepts of space and map are equally important and one can even look at a space as a map from this space onto a singleton space and in this manner identify these two concepts. With this in mind, a branch of General Topology which has become known as General Topology of Continuous Maps, or Fibrewise General Topology, was initiated. This field of research is concerned most of all in extending the main notions and results concerning topological spaces to those of continuous maps. In this way one can see some well-known results in a new and clearer light and one can also be led to further developments which otherwise would not have suggested themselves. The fibrewise viewpoint is standard in the theory of fibre bundles, however, it has been recognized relatively recently that the same viewpoint is also as important in other areas such as General Topology.peer-reviewe

Partial topological products in MAP

In this paper we continue with the study of the category MAP of continuous maps and their morphisms, introduced in [3]. This category is an extension of both the category TOPY (of continuous maps into a fixed space Y and their morphisms) and TOP (of topological spaces and continuous maps as morphisms). Partial products are used to obtain universal type theorems for T0, Tychonoff and zero-dimensional maps. Finally we introduce zero-dimensional and strongly zero-dimensional maps and generalize some well known results in the category TOP concerning zero-dimensional and strongly zero-dimensional spaces to the category MAP.peer-reviewe

Connective spaces

A new category of connective spaces is defined, which includes topological spaces and simple graphs, and generalizes the concept of connectedness. Not every connective space has a compatible topology; those that do are characterized by compatible partial orders.peer-reviewe

On superparacompact and Lindelof GO-spaces

In this paper we study some compact/paracompact type properties, namely weak superparacompactness, superparacompactness and Lindelofness. Particular attention is given to GO-spaces. It is proved that a GO-space X is weakly superparacompact if and only if every gap is a W- gap and every pseudogap is a W-pseudogap. A characterization of Lindelof GO-spaces involving C-(pseudo)gaps is given. We also show that there is a 1-l correspondence between superparacompact (resp. LindelGf) GO-d- extensions and preuniversal ODF (resp. prelindelsf) GO-uniformities. Finally we give several examples corresponding to the above results.peer-reviewe

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, U, ≤) (PO-quasi-uniform spaces) as those spaces with a biconvex quasi-uni- formity U on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology τU∗ of a PO-quasi-uniform space (X, U, ≤), the bi- completion (X, e Ue) of (X, U) is also a PO-quasi-uniform space (X, e Ue, ) with a partial order on Xe that extends ≤ in a natural way.peer-reviewe

On middle box products and paracompact cardinals

The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of many factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type ${}^{<\kappa}\square 2^\lambda$, where we prove that for a regular uncountable cardinal $\kappa$, if ${}^{<\kappa}\square 2^\lambda$ is paracompact for every $\lambda\ge\kappa$, then $\kappa$ is at least inaccessible. The case of the products of the type ${}^{<\kappa}\square X^\lambda$ for $\kappa$ singular has not been studied much in the literature and we offer various results. The question if ${}^{<\kappa}\square 2^\lambda$ can be paracompact for all $\lambda$ when $\kappa$ is singular has been partially answered but remains open in general.Comment: The version after the referee repor

On strong cellularity type properties of Lindelof groups

We prove several facts about cellularity and κ-cellularity of λ-Lindelöf groups generated by their κ-stable subspaces. For example, if a Lindelöf group G is generated by its κ-stable subspace then κ-cellularity (and hence cellularity) of G does not exceed κ. In particular, ω1-cellularity (and hence cellularity) of a Lindelöf group does not exceed ω1 if this group is generated by its ω1-Lindelöf subspace which is a P-space. For any cardinal μ with ω<μ c a Lindelöf group G is constructed which is separable (and hence has countable cellularity) while ω-cellularity of G is equal to μ.peer-reviewe