Serdica Mathematical Journal

Volume 24, Number 1, 1998

Originally, this volume was designed to be published in connection with the seventieth birthday of Professor Doitchin Bogdanov Doitchinov, the person who started regular readings in General Topology for Bulgarian students in the early sixties and greatly influenced the development of this subject in Bulgaria. Unfortunately, short before becoming 70, his health went down and on 29.11.1996, after a severe heart attack, he passed away. Missing him are his wife, his two sons, his former students in Topology and Analysis and many colleagues in Bulgaria and all over the world. This volume partially reflects the strong desire to pay tribute to his role in the Bulgarian mathematical life.

C O N T E N T S

• Doitchin Doitchinov - Biographical data (pp. iv-v)
• Doitchin Doitchinov - List of publications (pp. vi-viii)
• Császár, Á. On a class of D-Cauchy filters (pp. 1-4)
• Choban, M. Isomorphism problems for the Baire function spaces of topological spaces (pp. 5-20)
• Bentley, H. L., H. Herrlich Doitchinov's construct of supertopological spaces is topological (pp. 21-24)
• Trnková, V. Co-connected spaces (pp. 25-36)
• Lindgren, W. F., A. Szymanski Projectively solid sets and an n-dimensional Piccard's theorem (pp. 37-48)
• Kenderov, P. S., W. B. Moors, J. P. Revalski Dense continuity and selections of set-valued mappings (pp. 49-72)
• Künzi, H.-P. A., S. Romaguera, O. V. Sipacheva The Doitchinov completion of a regular paratopological group (pp. 73-88)
• Kocinac. L. R. D. p-sequential spaces and cleavability (pp. 89-94)
• P. Fletcher, P., W. Hunsaker A note on totally bounded quasi-uniformities (pp. 95-98)
• Dimov, G. Regular and other kinds of extensions of topological spaces (pp. 99-126)
• Simon, P., G. Tironi Pseudoradial spaces: finite products and an example from CH (pp. 127-134)

A B S T R A C T S

ON A CLASS OF D-CAUCHY FILTERS
Ákos Császár csaszar@lundens.elte.hu

1991 Mathematics Subject Classification: 54E15, 54D35. Key words: quasi-uniform space, D-Cauchy filter, firm extension.

In a quasi-uniform space, firmly D-Cauchy filters are introduced and they role in constructing firm extensions is investigated.

ISOMORPHISM PROBLEMS FOR THE BAIRE FUNCTION SPACES OF TOPOLOGICAL SPACES
Mitrofan M. Choban

1991 Mathematics Subject Classification: 46A50, 54A25, 54C35, 37N17. Key words: Baire complemented Banach space, Baire function, scattered space, Baire topology, D-set.

Let a compact Hausdorff space X contain a non-empty perfect subset. If a < b and b is a countable ordinal, then the Banach space B_a(X) of all bounded real-valued functions of Baire class a on X is a proper subspace of the Banach space B_b(X). In this paper it is shown that:

1. B_a(X) has a representation as C(b_aX), where b_aX is a compactification of the space PX - the underlying set of X in the Baire topology generated by the G_d-sets in X.

2. If 1 £ a < b £ W, where W is the first uncountable ordinal number, then B_a(X) is uncomplemented as a closed subspace of B_b(X).

These assertions for X = [0, 1] were proved by W. G. Bade [4] and in the case when X contains an uncountable compact metrizable space - by F.K.Dashiell [9]. Our argumentation is one non-metrizable modification of both Bade's and Dashiell's methods.

DOITCHINOV'S CONSTRUCT OF SUPERTOPOLOGICAL SPACES IS TOPOLOGICAL
H. L. Bentley fac1842@uoft01.utoledo.edu
Horst Herrlich herrlich@math.uni-bremen.de

1991 Mathematics Subject Classification: 54B30, 54A05, 54E05, 18D30. Key words: supertopological space, topological construct

It is shown that the construct of supertopological spaces and continuous maps is topological.

CO-CONNECTED SPACES
Vera Trnková

1991 Mathematics Subject Classification: 54C05, 08A40. Key words: products, continuous maps, monoids, connectednes.

Co-connected spaces, i.e. the spaces X for which any continuous map X²® X factors through a projection, are investigated. The main result: every free monoid is isomorphic to the monoid of all nonconstant continuous selfmaps of a metrizable co-connected space.

PROJECTIVELY SOLID SETS AND AN n-DIMENSIONAL PICCARD'S THEOREM
W. F. Lindgren william.lindgren@sru.edu
A. Szymanski andrzej.szymanski@sru.edu

1991 Mathematics Subject Classification: 54C10, 26B10, 22A15. Key words: Baire-open maps, semigroup, product.

We discuss functions f:X×Y® Z such that sets of the form f(A×B) have non-empty interiors provided that A and B are non-empty sets of second category and have the Baire property.

DENSE CONTINUITY AND SELECTIONS OF SET-VALUED MAPPINGS
Petar S. Kenderov pkend@math.bas.bg
Warren B. Moors moors@auckland.ac.nz
Julian P. Revalski revalski@math.bas.bg

1991 Mathematics Subject Classification: 54C60, 54C65. Key words: set-valued mappings, selections, semi-continuity, quasi-continuity, generic, Baire category.

A theorem proved by Fort in 1951 says that an upper or lower semi-continuous set-valued mapping from a Baire space A into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense G_d-subset of A.

In this paper we show that the conclusion of Fort's theorem holds under the weaker hypothesis of either upper or lower quasi-continuity. The existence of densely defined continuous selections for lower quasi-continuous mappings is also proved.

THE DOITCHINOV COMPLETION OF A REGULAR PARATOPOLOGICAL GROUP
Hans-Peter A. Künzi kunzi@math-stat.unibe.ch
Salvador Romaguera sromague@mat.upv.es
Ol'ga V. Sipacheva sipa@sipa.mccme.ru

1991 Mathematics Subject Classification: 54E15, 22A05. Key words: quasi-uniformity, quiet, Doitchinov complete, balanced, left K-complete, paratopological group.

We show that the two-sided quasi-uniformity U_B of a regular paratopological group (G,·) is quiet. The Doitchinov completion ([^G],[^(U_B)]) of (G,U_B) can be considered a paratopological group containing G as a doubly dense subgroup whenever G is Abelian. Furthermore [^(U_B)] is the two-sided quasi-uniformity of ([^G],·). These results generalize in an appropriate way important results about topological groups to regular paratopological groups. A counterexample dealing with the non-Abelian case is presented.

Furthermore we give conditions, depending on quasi-uniform completeness properties, under which a paratopological group is a topological group.

p-SEQUENTIAL SPACES AND CLEAVABILITY
Ljubisa R. D. Kocinac kocinac@archimed.filfak.ni.ac.yu

1991 Mathematics Subject Classification: 54A20, 54C05, 54D30, 54D55. Key words: p-sequential space, FU(p)-space, sequential space, cleavability, p-closed space, p-compact space.

We consider some relations between p-sequential-like properties and cleavability of topological spaces. Under a special assumption we give an very easy proof of the following result of A. V. Arhangel'skii (the main result in [1]): if a (countably) compact space X is cleavable over the class of sequential spaces, then X is also sequential.

A NOTE ON TOTALLY BOUNDED QUASI-UNIFORMITIES
P. Fletcher, W. Hunsaker

1991 Mathematics Subject Classification: 54E15. Key words: D-complete, quite.

We present the original proof, based on the Doitchinov completion, that a totally bounded quiet quasi-uniformity is a uniformity. The proof was obtained about ten years ago, but never published. In the meantime several stronger results have been obtained by more direct arguments [8,9,10]. In particular it follows from Künzi's [8] proofs that each totally bounded locally quiet quasi-uniform space is uniform, and recently Déak [10] observed that even each totally bounded Cauchy quasi-uniformity is a uniformity.

REGULAR AND OTHER KINDS OF EXTENSIONS OF TOPOLOGICAL SPACES
G. Dimov

1991 Mathematics Subject Classification: 54D35, 54E05, 54E17, 54D45. Key words: regular (regular-closed, compact, locally compact, completely regular, CE-regular) extensions; SR- (R-, RC-, EF-) proximities; nearness spaces; OCE- (CE-) regular spaces.

In this paper the notion of SR-proximity is introduced and in virtue of it some new proximity-type descriptions of the ordered sets of all (up to equivalence) regular, resp. completely regular, resp. locally compact extensions of a topological space are obtained. New proofs of the Smirnov Compactification Theorem [31] and of the Harris Theorem on regular-closed extensions [17, Thm. H] are given. It is shown that the notion of SR-proximity is a generalization of the notions of RC-proximity [17] and Efremovi c proximity [15]. Moreover, there is a natural way for coming to both these notions starting from the SR-proximities. A characterization (in the spirit of M. Lodato [23, 24]) of the proximity relations induced by the regular extensions is given. It is proved that the injectively ordered set of all (up to equivalence) regular extensions of X in which X is 2-combinatorially embedded has a largest element (kX,k). A construction of kX is proposed. A new class of regular spaces, called CE-regular spaces, is introduced; the class of all OCE-regular spaces of J. Porter and C. Votaw [29] (and, hence, the class of all regular-closed spaces) is its proper subclass. The CE-regular extensions of the regular spaces are studied. It is shown that SR-proximities can be interpreted as bases (or generators) of the subtopological regular nearness spaces of H. Bentley and H. Herrlich [4].

PSEUDORADIAL SPACES: FINITE PRODUCTS AND AN EXAMPLE FROM CH
Petr Simon psimon@ms.mff.cuni.cz
Gino Tironi tironi@univ.trieste.it

1991 Mathematics Subject Classification: 54G20, 54G15, 54D30, 54A20, 54B10. Key words: Radial space, pseudoradial space, compact space.

Aiming to solve some open problems concerning pseudoradial spaces, we shall present the following: Assuming CH, there are two semi-radial spaces without semi-radial product. A new property of pseudoradial spaces insuring the pseudoradiality of a product is presented.