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
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)
On a class of D-Cauchy filters (pp. 1-4)
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)
Co-connected spaces (pp. 25-36)
Lindgren, W. F., A. Szymanski
Projectively solid sets and an n-dimensional Piccard's theorem
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
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)
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
1991 Mathematics Subject Classification: 54E15, 54D35.
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.
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 Ba(X) of all bounded
real-valued functions of Baire class a on X is a proper
subspace of the Banach space Bb(X). In this paper it
is shown that:
1. Ba(X) has a representation as
where baX is a compactification of the space PX -
the underlying set of X in the Baire topology generated by the
Gd-sets in X.
2. If 1 £ a < b £ W, where W
is the first uncountable ordinal number, then
is uncomplemented as a closed subspace of
These assertions for X = [0, 1]
were proved by W. G. Bade 
and in the case when X contains an uncountable compact
metrizable space - by F.K.Dashiell . Our argumentation
is one non-metrizable modification of both Bade's and
DOITCHINOV'S CONSTRUCT OF SUPERTOPOLOGICAL SPACES IS TOPOLOGICAL
H. L. Bentley
1991 Mathematics Subject Classification: 54B30, 54A05, 54E05, 18D30.
supertopological space, topological construct
It is shown that the construct of supertopological spaces and continuous
maps is topological.
1991 Mathematics Subject Classification:
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
PROJECTIVELY SOLID SETS AND AN
n-DIMENSIONAL PICCARD'S THEOREM
W. F. Lindgren
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
non-empty sets of second category and have the Baire property.
AND SELECTIONS OF SET-VALUED MAPPINGS
Petar S. Kenderov
Warren B. Moors
Julian P. Revalski
1991 Mathematics Subject Classification: 54C60, 54C65.
Key words: set-valued
mappings, selections, semi-continuity, quasi-continuity, generic, Baire
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
Gd-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
THE DOITCHINOV COMPLETION OF A REGULAR PARATOPOLOGICAL GROUP
Hans-Peter A. Künzi
Ol'ga V. Sipacheva
1991 Mathematics Subject Classification: 54E15, 22A05.
quasi-uniformity, quiet, Doitchinov complete,
balanced, left K-complete, paratopological group.
We show that the two-sided quasi-uniformity UB of
a regular paratopological group (G,·) is quiet. The
Doitchinov completion ([^G],[^(UB)])
of (G,UB) can be considered a paratopological group
containing G as a doubly dense subgroup whenever G is Abelian.
Furthermore [^(UB)] 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
1991 Mathematics Subject Classification: 54A20, 54C05, 54D30, 54D55.
p-sequential space, FU(p)-space,
sequential space, cleavability, p-closed 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 ): 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
In particular it follows from Künzi's  proofs
that each totally bounded locally quiet quasi-uniform space is uniform, and
recently Déak  observed that even each totally bounded Cauchy
quasi-uniformity is a uniformity.
REGULAR AND OTHER KINDS OF EXTENSIONS OF TOPOLOGICAL SPACES
1991 Mathematics Subject Classification: 54D35, 54E05, 54E17, 54D45.
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  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
 and Efremovi c proximity .
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
 (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 .
PSEUDORADIAL SPACES: FINITE PRODUCTS AND AN EXAMPLE FROM
1991 Mathematics Subject Classification: 54G20, 54G15, 54D30,
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.