Serdica Mathematical Journal

Volume 44, Number 1−2, 2018

This issue of Serdica Mathematical Journal is dedicated
to the memory of Professor Stoyan Nedev (1942−2015).

Guest Editors: V. Valov, V. Gutev

C O N T E N T S

A B S T R A C T S

PROFESSOR STOYAN YORDANOV NEDEV
Alexander V. Arhangel’skii arhangel.alex@gmail.com,
Mitrofan M. Choban mmchoban@gmail.com,
Petar S. Kenderov vorednek@yahoo.com

Professor Stoyan Yordanov Nedev, age 72, died in Sofia, Bulgaria, in the early morning hours of Sunday, May 31, 2015. He was survived by his wife Kalinka, son Yordan and doughter Ivelina. Stoyan Nedev was a reliable friend, a great father, an excellent husband and a gifted mathematician. In this article, we give a a short outline of his life, discuss some of his mathematical contributions and present a list of his major publications.

TOPOLOGICAL REPRESENTATION OF PRECONTACT ALGEBRAS AND A CONNECTED VERSION OF THE STONE DUALITY THEOREM − II
Georgi Dimov gdimov@fmi.uni-sofia.bg,
Dimiter Vakarelov dvak@fmi.uni-sofia.bg

2010 Mathematics Subject Classification: 54E05, 54H10, 54D80, 06E15, 03G05, 54D30, 54D10, 54D05, 54D35, 54G05, 54C25.
Key words: (pre)contact algebra, 2-(pre)contact space, 3-(pre)contact space, (C-)semiregular space, (C-)weakly regular space, (C)N-regular space, (C-)regular space, compact Hausdorff space, compact T₀-extension, Stone space, (complete) Boolean algebra, Stone adjacency space, absolute, 2-combinatorial embedding, open combinatorial embedding.

The notions of extensional (and other kinds) 3-precontact and 3-contact spaces are introduced. Using them, new representation theorems for precontact and contact algebras, satisfying some additional axioms, are proved. They incorporate and strengthen both the discrete and topological representation theorems from [11, 6, 7, 12, 22]. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. In particular, such a bijective correspondence for the RCC systems of [19] is obtained, strengthening in this way the previous representation theorems from [12, 6]. As applications of the obtained results, we prove several Smirnov-type theorems for different kinds of compact semiregular T₀-extensions of compact Hausdorff extremally disconnected spaces. Also, for every compact Hausdorff space X, we construct a compact semiregular T₀-extension (ϰX, ϰ) of X which is characterized as the unique, up to equivalence, C-semiregular extension (cX,c) of X such that c(X) is 2-combinatorially embedded in cX; moreover, ϰ X contains as a dense subspace the absolute EX of X.

SELECTION PRINCIPLES IN (3, 1, ρ)-D-METRIZABLE SPACES AND (3, 2, ρ)-D-METRIZABLE SPACES
Tomi Dimovski tomi.dimovski@gmail.com,
Dončo Dimovski donco@pmf.ukim.mk

2010 Mathematics Subject Classification: Primary 54D20, Secondary 54A05, 54E99.
Key words: (3, 1, ρ)-D-metrizable spaces, (3, 2, ρ)-D-metrizable spaces, M-bounded space, R-bounded space, H-bounded space, D-precompact space, σ-D-precompact space, D-pre-Lindelof space, D-bounded space.

In this paper we introduce and investigate M, R and H-boundedness properties in (3, j, ρ)-D-metrizable spaces, j ∈ {1,2}, related to the classical covering properties of Menger, Rothberger and Hurewicz.

HOMOLOGICAL DIMENSION AND DIMENSIONAL FULL-VALUEDNESS
Vesko Valov veskov@nipissingu.ca

2010 Mathematics Subject Classification: Primary 55M10, 55M15; Secondary 54F45, 54C55.
Key words: homological dimension, homology groups, homogeneous metric ANR-compacta.

There are different definitions of homological dimension of metric compacta involving either Čech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X with respect to any field coincide provided X is homologically locally connected with respect to the singular homology up to dimension n = dim X (br., X is lcⁿ). We also prove that any two-dimensional lc² metric compactum X satisfies the equality dim (X × Y) = dim X + dim Y for any metric compactum Y (i.e., X is dimensionally full-valued). This improves the well known result of Kodama [6, Theorem 8] that every two-dimensional ANR is dimensionally full-valued. Actually, the condition X to be lc² can be weaken to the existence at every point x ∈ X of a neighborhood V of x such that the inclusion homomorphism H_k(V̅; S¹) → H_k(X; S¹) is trivial for all k = 1, 2.

VIETORIS-TYPE TOPOLOGIES ON HYPERSPACES
Elza Ivanova-Dimova elza@fmi.uni-sofia.bg

2010 Mathematics Subject Classification: 54B20, 54D10, 54C05.
Key words: hyperspace, Vietoris (hyper)topology, Vietoris-type (lower-Vietoris-type, upper-Vietoris-type, Tychonoff-type) (hyper)topology, (ℱ, 𝒢)-hit-and-miss topology.

We study the (ℱ, 𝒢)-hit-and-miss topology introduced by Clementino and Tholen [4]. The same topology was introduced independently in the preliminary version [12] of this paper under the name of Vietoris-type hypertopology (at that time we were not aware of the paper [4]). We show that the Vietoris-type hypertopology is, in general, different from the Vietoris topology. Also, some of the results of E. Michael [13] about hyperspaces with Vietoris topology are extended to analogous results for hyperspaces with Vietoris-type topology. We obtain as well some results about hyperspaces with Vietoris-type topology which concern some problems analogous to those regarded by H.-J. Schmidt in [14].

PRODUCT OF PROPER SHAPE EQUIVALENCES OVER FINITE COVERINGS IS AN EQUIVALENCE
Nikita Shekutkovski nikita@pmf.ukim.mk,
Abdulla Buklla abdullabuklla@hotmail.com

2010 Mathematics Subject Classification: 54C56.
Key words: Product of morphisms, proper proximate sequence over finite coverings, intrinsic shape, finite covering, shape of product.

In this paper we prove that in the category of proper shape over finite coverings from Sh_pF(X) = Sh_pF(X') and Sh_pF(Y) = Sh_pF(Y') it follows that Sh_pF(X × Y) = Sh_pF(X' × Y'). Also, we give an example in which the product of two morphisms is not morphism in the category of proper shape.

CONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES OF COVERS
Valentin Gutev valentin.gutev@um.edu.mt

2010 Mathematics Subject Classification: 54C60, 54C65, 54F35, 54F45.
Key words: Lower semi-continuous mapping, lower locally constant mapping, continuous selection, local connectedness in finite dimension, finite-dimensional space.

It is given a simplified and self-contained proof of the classical Michael's finite-dimensional selection theorem. The proof is based on approximate selections constructed stepwise over skeletons of nerves of covers. The method is also applied to simplify the proof of the Schepin−Brodsky's generalisation of this theorem.

LOWER SEPARATION AXIOMS VIA BOREL AND BAIRE ALGEBRAS
Taras Banakh t.o.banakh@gmail.com,
Adam Bartoš drekin@gmail.com

2010 Mathematics Subject Classification: Primary 54D10; Secondary 54H05, 54B05, 54B10.
Key words: Separation axiom, Borel set, Baire property, nowhere dense set.

Let κ be an infinite regular cardinal. We define a topological space X to be a T_κ-Borel-space (resp. a T_κ-BP-space) if for every x ∈ X the singleton {x} belongs to the smallest κ-additive algebra of subsets of X that contains all open sets (and all nowhere dense sets) in X. Each T₁-space is a T_κ-Borel-space and each T_κ-Borel-space is a T₀-space. On the other hand, T_κ-BP-spaces need not be T₀-spaces.
We prove that a topological space X is a T_κ-Borel-space (resp. a T_κ-BP-space) if and only if for each point x ∈ X the singleton {x} is the intersection of a closed set and a G_κ-set in X (resp. {x} is either nowhere dense or a G_κ-set in X). Also we present simple examples distinguishing the separation axioms T_κ-Borel and T_κ-BP for various infinite cardinals κ, and we relate the axioms to several known notions, which results in a quite regular two-dimensional diagram of lower separation axioms.

ON MONOTONE ORTHOCOMPACTNESS
Strashimir G. Popvassilev strash.pop@gmail.com,
John E. Porter jporter@murraystate.edu

2010 Mathematics Subject Classification: 54D20, 54D70, 54F05, 54B10, 54G20, 54D65.
Key words: monotone covering properties, monotonically orthocompact, orthobase, transitive neighbornet, GO-space, regressive functions on $\omega_1$, NSR base.

Junnila and Künzi defined monotone orthocompactness via transitive neighbornets, and proved that monotonically normal, monotonically orthocompact spaces must have an ortho-base. Answering one of Junnila and Künzi's questions, Shouli and Yuming claimed to have provided an example of a monotonically orthocompact space without an ortho-base. We define a version of monotone orthocompactness via interior-preserving open refinements and show that it is a strictly weaker property than monotone orthocompactness of Junnila and Künzi, and we point out an error in the paper by Shouli and Yuming, thereby indicating that the question of Junnila and Künzi appears to remain open.

EQUIVARIANT ABSOLUTE EXTENSORS FOR FREE ACTIONS OF COMPACT GROUPS
Sergey Ageev ageev@bsu.by,
Alexander Dranishnikov dranish@math.ufl.edu,
James Keesling kees@math.ufl.edu

2010 Mathematics Subject Classification: Primary 55M15, 57S10; Secondary 55R35.
Key words: compact group, absolute extensor.

For every compact metrizable group G there is a free universal G-action on the Hilbert space ℓ₂ which makes ℓ₂ a G-equivariant absolute extensor for the class of free G-spaces.

CARDINAL INEQUALITIES FOR URYSOHN SPACES INVOLVING VARIATIONS OF THE ALMOST LINDELÖF DEGREE
Ivan S. Gotchev gotchevi@ccsu.edu

2010 Mathematics Subject Classification: Primary 54A25; Secondary 54D10, 54D20.
Key words: Cardinal function, almost Lindelöf degree, θ-closure, θ-tightness, θ-bitightness, θ²-pseudocharacter.

Recall that for a topological space X, t_&\theta;1(X) is the smallest infinite cardinal κ such that for every A ⊂ X and every x ∈ cl(A) there exists a set B ⊂ A such that |B| ≤ κ and x ∈ cl_θ(B) ([6]). For every Urysohn space X we define the cardinal function ψ_θ²(X), the θ²-pseudocharacter of   X, as the smallest infinite cardinal κ such that for each x ∈ X, there is a collection 𝒱_x of open neighborhoods of x such that |𝒱_x| ≤ κ and ∩{cl_θ(cl(V)) : V ∈ 𝒱_x} = {x}. Using this new cardinal function, among other results, we show that if X is a Urysohn space and A ⊂ X then (1) |cl(A)| ≤ |A|^{t_θ1(X)ψ_θ2(X)}; and (2) |X| ≤ 2^{t_θ1(X)ψ_θ2(X)aL_c(X)}.
Since for every Urysohn space X we have ψ_θ²(X) ≤ ψ(X)L(X), inequality (2) sharpens, for the class of Urysohn spaces, the famous Arhangel'skiï−Šapirovskiï inequality |X| ≤ 2^{t(X)ψ(X)L(X)}, which is valid for every Hausdorff space X.
Using the cardinal function bt_θ, called θ-bitightness, or recently introduced in [3] variations of the almost Lindelöf degree, other upper bounds of the cardinality of Urysohn spaces are proved which improve (2), Kočinac' inequality |X| ≤ 2^{bt_θ(X)aL(X)}, which is valid for every Urysohn space X, and, for special classes of Urysohn spaces, Bella-Cammaroto's inequality |X| ≤ 2^{t(X)ψ_c(X)aL_c(X)}, which is true for every Hausdorff space X.

FRAGMENTABILITY OF OPEN SETS VIA SET-VALUED MAPPINGS
Mitrofan M. Choban mmchoban@gmail.com,
Petar S. Kenderov kenderovp@cc.bas.bg,
Julian P. Revalski revalski@math.bas.bg

2010 Mathematics Subject Classification: 54E52, 54C60, 49J27, 49J53.
Key words: fragmentability of (open) sets, set-valued mappings, topological game, winning strategy, Baire category, strong minimum.

We study the notion of fragmentability of nonempty open sets in a topological space X and provide several characterizations of this concept via properties of set-valued mappings taking their values in X. As a corollary we obtain that in a compact space X the nonempty open sets are fragmentable if, and only if, the set of all continuous real-valued functions in X which attain their minimum at exactly one point in X is a residual subset of the space C(X) of all bounded continuous real-valued functions in X, equipped with the uniform convergence norm.

MORE ON THE CARDINALITY OF S(n)-SPACES
Ivan S. Gotchev gotchevi@ccsu.edu,
Ljubiša D. R. Kočinac lkocinac@gmail.com

2010 Mathematics Subject Classification: Primary 54A25, 54D10.
Key words: Cardinal function, S(n)-space, S(n)-density, S(n)-discrete, S(n)-pseudocharacter, S(n)-tightness, S(n)-bitightness.

In this paper, for a topological space X and any positive integer n, we define the cardinal functions d_n(X), t_n(X) and bt_n(X), called respectively S(n)-density, S(n)-tightness and S(n)-bitightness, and using them and recently introduced in [10] cardinal functions χ_n(X), ψ_n(X), and s_n(X), called respectively S(n)-character, S(n)-pseudocharacter, and S(n)-spread, we prove some cardinal inequalities for S(n)-spaces, which extend to the class of S(n)-spaces some results of Pospišil, Arhangel'skiĭ, Hajnal and Juhász, Shapirovskiĭ and Kočinac. Two representative results are: If X is an S(n)-space, then |X| ≤ 2^2d_n(X) and |X| ≤ [d_n(X)]^bt_n(X).