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].     

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.     

Michael's problem and weakly infinite-dimensional spaces

Let X be a compact Hausdorff space. Suppose that any multivalued map , where Y is a Gδ subset of a Banach space, such that the values of F are convex and closed in Y, has a continuous single-valued selection. Then we prove that X is weakly infinite-dimensional. This provides a partial solution of Gδ-problem, posed by Ernest Michael.     

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.     

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.     

On closedness assumptions in selection theorems

We begin by a short survey of various attempts in selection theory to avoid the closedness assumption for values of multivalued mappings. We collect special cases when Michael's Gδ-problem admits an affirmative solution and we prove some unified theorems of such type. We also show that in general this problem has a negative solution. In comparison with a recent result of Filippov, we work directly in the Hilbert cube rather than in the space of all probabilistic measures endowed with different topologies.     

Variations on selective separability in non-regular spaces

We study various variations on selective separability in non-regular topological spaces. We use the notions of θ-closure and θ-density to define selective versions of θ-separability. These properties are also related to topological games.     

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.     

Strong paracompactness and multi-selections

We characterize strong paracompactness in terms of usco multi-selections for closed-valued lower semi-continuous mappings into completely metrizable spaces, thus generalizing recent results obtained by Choban, Mihaylova and Nedev [M. Choban, E. Mihaylova, S. Nedev, On selections and classes of spaces, Topology Appl. 155 (2008) 797-804]. Related results and applications are achieved as well.     

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.     

Covering dimension and finite-to-one maps

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 n²-to-1 map.     

Extension dimension and quasi-finite CW-complexes

We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define UV(L)-spaces in the realm of metrizable spaces and show that some properties of UV(n)-spaces and UV(n)-maps remain valid for UV(L)-spaces and UV(L)-maps, respectively.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Extraordinary dimension of maps

We consider the extraordinary dimension dimL introduced recently by Shchepin [E.V. Shchepin, Arithmetic of dimension theory, Russian Math. Surveys 53 (5) (1998) 975-1069]. If L is a CW-complex and X a metrizable space, then dimLX is the smallest number n such that ΣnL is an absolute extensor for X, where ΣnL is the nth suspension of L. We also write dimLf?n, where is a given map, provided dimLf⁻¹(y)?n for every y∈Y. The following result is established: Supposeis a perfect surjection between metrizable spaces, Y a C-space and L a countable CW-complex. Then conditions (1)-(3) below are equivalent:

(1)

dimLf?n;

(2)

There exists a dense andGδsubsetGofC(X,In)with the source limitation topology such thatdimL(f×g)=0for everyg∈G;

(3)

There exists a mapis such thatdimL(f×g)=0;If, in addition, X is compact, then each of the above three conditions is equivalent to the following one;

(4)

There exists anFσsetA⊂Xsuch thatdimLA?n−1and the restriction mapf|(X?A)is of dimensiondimf|(X?A)?0.

     

Fiberwise absolute neighborhood extensors for a class of metrizable spaces

In the paper we study fiberwise absolute neighborhood extensors with respect to some classes of metrizable spaces by means of the local extension properties and the lifting properties of the underlying spaces.     

Bead spaces and fixed point theorems

The notion of a bead metric space defined here (see Definition 6) is a nice generalization of that of the uniformly convex normed space. In turn, the idea of a central point for a mapping when combined with the “single central point” property of the bead spaces enables us to obtain strong and elegant extensions of the Browder-Göhde-Kirk fixed point theorem for nonexpansive mappings (see Theorems 14-17). Their proofs are based on a very simple reasoning. We also prove two theorems on continuous selections for metric and Hilbert spaces. They are followed by fixed point theorems of Schauder type. In the final part we obtain a result on nonempty intersection.     

Lindelöf spaces which are indestructible, productive, or D

We discuss relationships in Lindelöf spaces among the properties “indestructible”, “productive”, “D”, and related properties.     

The colorings of homeomorphisms on connected graphs

In this paper, we decide the exact value of the color number of a fixed point free homeomorphism on a connected locally finite graph. We prove that for every fixed-point free homeomorphism from a connected locally finite graph into itself, the greatest common divisor of all period for its map is equal to one or three if and only if its color number is 4.