Versions of separability in bitopological spaces

We study selective versions of separability in bitopological spaces. In particular, we investigate these properties in function spaces endowed with the topology of pointwise convergence and the compact-open topology.     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².     

Very I-favorable spaces

We prove that a Hausdorff space X is very I-favorable if and only if X is the almost limit space of a σ-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very I-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.     

Game approach to universally Kuratowski-Ulam spaces

We consider a version of the open-open game, indicating its connections with universally Kuratowski-Ulam spaces. From [P. Daniels, K. Kunen, H. Zhou, On the open-open game, Fund. Math. 145 (3) (1994) 205-220] and [D. Fremlin, T. Natkaniec, I. Rec?aw, Universally Kuratowski-Ulam spaces, Fund. Math. 165 (3) (2000) 239-247] topological arguments are extracted to show that: Every I-favorable space is universally Kuratowski-Ulam, Theorem 8; If a compact space Y is I-favorable, then the hyperspaceexp(Y)with the Vietoris topology is I-favorable, and hence universally Kuratowski-Ulam, Theorems 6 and 9. Notions of uK-U and uK-U^∗ spaces are compared.     

Maps and f-normal spaces

It has long been known that hyper-real maps preserve realcompactness. In this paper we show that hyper-real maps preserve nearly realcompactness as well. We will also introduce the concepts of ε-perfect maps and f-normal spaces and explore them in a way that mirrors Rayburn's 1978 study of δ-perfect maps and h-normal spaces.     

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to _ω1${\omega }_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “_ℵ1${\aleph }_1$-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.     

Minimal KC spaces are compact

We show that every KC space (X,τ), such that τ is minimal among the KC topologies on X, must be compact (not necessarily T₂). This solves a long-standing question, first raised by R. Larson in 1973.     

A new approach to fixed point theorems on G-metric spaces

In this paper, we introduce the metric d^G$d^G$ on a G  -metric space (X,G)$(X,G)$ and use this notion to show that many contraction conditions for maps on the G  -metric space (X,G)$(X,G)$ reduce to certain contraction conditions for maps on the metric space (X,d^G)$(X,d^G)$. As applications, the proofs of many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ may be simplified, and many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ are direct consequences of preceding results for maps on the metric space (X,d^G)$(X,d^G)$.     

Two results on spaces with a sharp base

Example. There exists a space X with a sharp base and a perfect mapping onto a space Y which does not have a sharp base.     

Selectively c.c.c. spaces

We present a study about a natural way of defining a selective version of the c.c.c. property. This definition and some related properties were already considered under different names in other works, such as Daniels et al. (1994) [9], Scheepers (2000) [12]. Here we will present some of its relations with other selective properties and we present some examples that show the differences among the properties considered. We also study the behavior of these properties when the products are considered.     

Realcompactness in maximal and submaximal spaces

We study realcompactness in the classes of submaximal and maximal spaces. It is shown that a normal submaximal space of cardinality less than the first measurable is realcompact. ZFC examples of submaximal not realcompact and maximal not realcompact spaces are constructed. These examples answer questions posed in [O.T. Alas, M. Sanchis, M.G. Tka?enko, V.V. Tkachuk, R.G. Wilson, Irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new ZFC examples, Topology Appl. 107 (3) (2000) 259-273] and generalize some results from [D.P. Baturov, On perfectly normal dense subspaces of products, Topology Appl. 154 (2) (2007) 374-383].     

Dually discrete spaces

A neighbourhood assignment in a space X is a family of open subsets of X such that x∈Ox for any x∈X. A set Y⊆X is a kernel ofO if . We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371-377].     

Moscow spaces and selection theory

A well-known result on Moscow spaces states that every Gδ-dense subset of a Moscow space X is C-embedded in X. We present here the selection version of this result and also (by means of two different approaches) we use selection theory to characterize the open bounded subsets of a uniform space (X,U) in the case when its completion is a Moscow space.     

Semitopological groups,Bouziad spaces and topological groups

A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we use topological games to show that many semitopological groups are in fact topological groups.     

Topological groups and covering dimension

We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional topological groups. The dimension function that we introduce extends Lebesgue covering dimension, has the hereditary property, and has a product theory that is more similar to the product theory for the finite dimensional case.     

The Baire property in hit-and-miss hypertopologies

Recently, some techniques have been developed for the study of the Baire property in hyperspaces. These techniques have been applied to solve a long-standing open problem of McCoy in 1975 and a recent open problem of Zsilinszky. In this paper, we extend and apply these techniques further to investigate the Baire property of hyperspaces equipped with the general hit-and-miss topology.     

Products of certain dually discrete spaces

If X and Y are locally compact GO spaces then X×Y is dually discrete. If μ and ν are two ordinals and X is a normal subspace of μ×ν then X is dually discrete.     

Lebesgue and co-Lebesgue di-uniform texture spaces

The author introduces the notions of Lebesgue di-uniformity and co Lebesgue di-uniformity and investigates the relationship between a Lebesgue quasi uniformity on X and the corresponding Lebesgue di-uniformity on the discrete texture (X,P(X)). Finally a notion of a dual dicovering Lebesgue quasi di-uniform texture space is introduced and several properties are discussed.