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. ivv)

Doitchin Doitchinov  List of publications (pp. viviii)

Császár, Á.
On a class of DCauchy filters (pp. 14)

Choban, M.
Isomorphism problems for the Baire function spaces
of topological spaces (pp. 520)

Bentley, H. L., H. Herrlich
Doitchinov's construct of supertopological spaces is topological (pp. 2124)

Trnková, V.
Coconnected spaces (pp. 2536)

Lindgren, W. F., A. Szymanski
Projectively solid sets and an ndimensional Piccard's theorem
(pp. 3748)

Kenderov, P. S., W. B. Moors, J. P. Revalski
Dense continuity and selections of setvalued mappings (pp. 4972)

Künzi, H.P. A., S. Romaguera, O. V. Sipacheva
The Doitchinov completion of a regular paratopological group
(pp. 7388)

Kocinac. L. R. D.
psequential spaces and cleavability (pp. 8994)

P. Fletcher, P., W. Hunsaker
A note on totally bounded quasiuniformities (pp. 9598)

Dimov, G.
Regular and other kinds of extensions of topological spaces (pp. 99126)

Simon, P., G. Tironi
Pseudoradial spaces: finite products and an example from CH (pp. 127134)
A B S T R A C T S
ON A CLASS OF DCAUCHY FILTERS
Ákos Császár
csaszar@lundens.elte.hu
1991 Mathematics Subject Classification: 54E15, 54D35.
Key words:
quasiuniform space, DCauchy filter, firm extension.
In a quasiuniform space, firmly DCauchy
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, Dset.
Let a compact Hausdorff space X contain a nonempty
perfect subset. If a < b and
b is a countable
ordinal, then the Banach space B_{a}(X) of all bounded
realvalued 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_{a}X),
where b_{a}X 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 nonmetrizable 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.unibremen.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.
COCONNECTED SPACES
Vera Trnková
1991 Mathematics Subject Classification:
54C05, 08A40.
Key words:
products, continuous maps, monoids, connectednes.
Coconnected spaces, i.e. the spaces X for which any
continuous map X^{2}® 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
coconnected space.
PROJECTIVELY SOLID SETS AND AN
nDIMENSIONAL 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: Baireopen maps, semigroup, product.
We discuss functions f:X×Y® Z such that sets of the form
f(A×B) have nonempty interiors provided that A and
B are
nonempty sets of second category and have the Baire property.
DENSE CONTINUITY
AND SELECTIONS OF SETVALUED 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: setvalued
mappings, selections, semicontinuity, quasicontinuity, generic, Baire
category.
A theorem proved by Fort in 1951 says that an upper or lower
semicontinuous setvalued mapping from a Baire space A
into nonempty compact subsets of a metric space is both
lower and upper semicontinuous 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
quasicontinuity. The existence of densely defined
continuous selections for lower quasicontinuous mappings is
also proved.
THE DOITCHINOV COMPLETION OF A REGULAR PARATOPOLOGICAL GROUP
HansPeter A. Künzi
kunzi@mathstat.unibe.ch
Salvador Romaguera
sromague@mat.upv.es
Ol'ga V. Sipacheva
sipa@sipa.mccme.ru
1991 Mathematics Subject Classification: 54E15, 22A05.
Key words:
quasiuniformity, quiet, Doitchinov complete,
balanced, left Kcomplete, paratopological group.
We show that the twosided quasiuniformity 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 twosided quasiuniformity of ([^G],·).
These results generalize in an appropriate way important
results about topological groups to regular paratopological groups.
A counterexample dealing with the nonAbelian case is presented.
Furthermore we give conditions, depending on quasiuniform completeness
properties, under which a paratopological group is a topological group.
pSEQUENTIAL SPACES AND CLEAVABILITY
Ljubisa R. D. Kocinac
kocinac@archimed.filfak.ni.ac.yu
1991 Mathematics Subject Classification: 54A20, 54C05, 54D30, 54D55.
Key words:
psequential space, FU(p)space,
sequential space, cleavability, pclosed space,
pcompact space.
We consider some relations between psequentiallike
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 QUASIUNIFORMITIES
P. Fletcher, W. Hunsaker
1991 Mathematics Subject Classification: 54E15.
Key words: Dcomplete, quite.
We present the original proof, based on the Doitchinov
completion, that a totally bounded quiet quasiuniformity 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 quasiuniform space is uniform, and
recently Déak [10] observed that even each totally bounded Cauchy
quasiuniformity 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 (regularclosed, compact, locally compact, completely
regular, CEregular) extensions; SR (R, RC, EF) proximities;
nearness spaces; OCE (CE) regular spaces.
In this paper the notion of SRproximity
is introduced and in virtue of it some new proximitytype
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 regularclosed extensions
[17, Thm. H] are given.
It is shown that the notion of SRproximity is a generalization of
the notions of RCproximity
[17] and Efremovi c proximity [15].
Moreover, there is a natural way for coming to both these notions
starting from the SRproximities.
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 2combinatorially embedded has a largest element (kX,k).
A construction of kX is proposed.
A new class of regular spaces, called CEregular spaces, is introduced;
the class of all OCEregular spaces of J. Porter and C. Votaw
[29] (and, hence, the class of all regularclosed spaces) is its
proper subclass. The CEregular extensions of the regular spaces
are studied. It is shown that SRproximities 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 semiradial spaces without
semiradial product. A new property of pseudoradial spaces insuring
the pseudoradiality of a product is presented.
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