SCIENTIFIC ABSTRACT SMIRNOV, YU.M. - SMIRNOV, YU.V.

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• Collection: Scientific Abstracts
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• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: August 24, 2000
• Sequence Number: 5
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

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Depictions of systems of open aggregates. Mat. sbor., 31, No. 1, 1952.
Monthly Lj�j of Rusnian Accessions, Library of Congress October 1952 UNCLASSIFIED

USSR/Mathematics - Modern Algebra, Jul/Aug 52
Reflections
"Reflections of Systems of Closed Sets," Yu. M.
Smirnov, Moscow
"Matemat Sbor" Vol XXXI (73), No 1, PP 152-166
Poses the problem of characterizing those homo-
morphisms Y in X which are generated by con-
tinuous reflections X in Y, and of constructing
according to a given homomorphism a continuous
reflection generating it. The aim of current
article is to solve this problem for a completely
22OT81
regular space X and bicompactum Y. The results
obtained are applied further to the problem of
continuing continuous ref lections v-ith a gi-ven
space into its given extension and to the prololem
concerning the difference of bicompact extensions
of one and the same space, Submitted 5 Mar 52.
2P-OTB1

S11MOV Yu. M.
PA 237T85
USSR/Mathematics - Topology Nov/Dec 52
"Proximity Spaces," Yu. M. Smirnov, Moscow
"Matemat Sbor" Vol 31 (73)) No 3., PP 543-574
Systematically investigates proximity spaces and
compares them with ordinary topological spaces.
Cites similar works (proximity, infinitesimal
spaces) of V. A. Yefremovich (1951-52), N. S.
Ramm (1951),, P. S. Aleksandrov (1950). P. S,
Uryson (1950), and A. D. Taymanov (1952).
237T85

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TO I I(A Of-V
./Smiloov, I'll. nil ilroximity npn~es fit fit(- spnse of-'V. A.
Ffivinovii'. I),J,h,I%- Mad. Naol~ SSSR (N k;J 8-1,89S
'ill., a tit litir pi oximity sparr-s [c f- v.g., (lie paper
-d abovvj and 11wir relations with uniforto spaces.
NI.mv wFiflis. art- given withmit ptotlfs.
A pioximily -;p;u-v (briefly, 6-slmre) R iq (-;illrfl roaximal
if it i:; clo,wd in ony A-spare '5-)R. It is shown timl a A-space
j,i mix6mal if and onk, if it is romparl (;1,; .1 topologiral
qmcq.); ex-cry A-spare may be illilleddcd (ill all essentially
lini(Ine way) in a unaximal 5-spacc; if R is a given completely
tcgohr topological spacv, then flivre is a onvAn-oilf- corie-
-.1101111PIRT boween poximily strurfurer oil R compatible
with it.-; topology and crimpact spaces containing R topo-
Ingically as a dense ~--Ilbwt.
For a Vivco 6-.-parc R consider the collection (partially
clolvi-rd in an obvions way) of all intifni-mitics compatible
with its pro-dini(Y strin1mv.. 11 is stated that this collection
hasa mininnim; 4 R N nif-frizable, iL hasa maximum. If such
A Invixintal onifol mity dxists but (Ines not arlinit of ;in exten-
sinn onto a 6-5--Parc SDR, SF~R, S=R, then R is called
complete. It is~fated.thatatiyb-spacc for which thereexists
;I Maximal imiformity may be imbedded in"a complete
It is to be noted that Theorem It of the artirle is not
,,,rri,t (.t-; pointed oui Ily the anthorin another note Esanic
Doklady (N.S.) 88, 761-704 (1953). last footnote oil p.
767-1). Af. Katitar (Pragne).

SMIRNOV, Yu.
Theory of completel.- reguJar spaces. Uch.zap.Mosk.un. no.155:137-
155 152- (Spaces, G--neralized) (YJRA 8:71)

liiathemiati~~s - laq~cloi
gy, licitliborhooJ
!ul/A7u' ~3
116P-ca.etry o&* 'ieigh-borhrods, Uniform. Geor.Ietry, anti a mm ~, r d S4 . S hv EL rs ,
ivar.ovo ~"tate Pedai,,or Inst
Ilat Sour, Vol 'JD (7-5), No 1. !)P 157-180
Cxjitinuatiori of V. A. Yeremovich's work ("Non-equivalepce of -Euclide~an mid Loba-
C`ievskil-n Sp-af-es'll L-721) Wit Nauk, VC1 Lw, No 2 (30), 1949). Demonstrate almost all --f the
results of Yu. i-I. ~i.,drnov's work ("Sp,-ices of !~-'eirhborhoods," Kat Sbor, Vol 31 (73),
19~121) I-.- ot.,.er, often si:.".,-,Ier, Purther, investipate the interconnect of a
numb r of infinitesIrmal concetAs wW~ the concept of neiehborhood i)ermits one to
3-Implify consi&~rably Mile proof of the princ-iral. tbeoremis of biconract exten21cns.
Preserted 17 Set) -J2".
271T86

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Smirnov, Yu. On the completem ~:~; o1 1)roximity spn( v3,
I Ak.vl. i\faiil; (N.S.) 88, 70-70-1 (111).-3).'
It l' i -; t'wNillill %- '~Jmce' thl-11 a I lillvt lioll t of-ralbsi"'; of,
P i,; vallol ;% , if, f(-r every imiform 6-covering [cf.
Smi,liov, 70.!- N.S. 31(173), 5.13-574 (1952); these
1-1. 1107-1 V,
Tll- t 'millet 14 m )f P is dellned. essi-nti illy, as the space ePi
III'l \Irll,ll cen I (-rel I(A-SN-st ellis [for 6-sy5terils cf.
liffiriv, ~-Inw I),,kl.idv (N.S.) 84, 893-808 (1952); these.
R-v. 14. 1101-11: P i~z calk-d complete if cP=P If P Q are:
pr(owilitv :p;!, thril a mapping f of P into is called

c-inapping if the image of anycenLred c-sN .stein is a c-system..
A proximity space is called totally bounded if any uniforms
6-covering contains a finite subcovering.
*rhe author gives (without proofs) many interestingi
theorems concerning completion, c-mappings, and totall
botindLdness of proximity spaces. Some characteristic re-
stilts: (1) cP is the largest of-all a-extensions R of P (i.e.,
proximity spaces containing P as a dense proximity sub-
space) such that every uniform 6-covering of P admits of an
extension to a uniform B-covering of R; (2) P is complete
if and only if every centred c-system of closed sets has a
non-void intersection; (3) a mapping f of P into Q may be
extended to a continuous mapping of cP into cQ if and only -
Af f is a c-mapping; (4) each of the following properties is,
equivalent with the total boundedness of P: a cP is
compact; (b) every real-valued c-function on P is nded;
(c) every psetidonictric in P is bounded; (b) the exists:
only one uniformitycompatible with the proximity structured'
Of 11. An exainple is given of a proximity space possessing
no inaximal (finest) uniformity- 111. Kat~i(w Wraclue).

WIRNOVI TU.M.
dompleteness of proximity spaces. Trudy Monk.imt.ob-T& 3:271-306
154. (KLU 7:7)
(Spaces. Generalized)
"r.

SMIRNOV, Yu.M.
Dimension of proximity spaces. Dokl.AN SSSR 95 no.4:717-720
Ap f54. (XLRA 7:3)
(Spacq4, Generalized)

SUBJECT USSR/MATHEMATICS/Topology CARD 1/2 PG - 15
AUTHOR SMIRNOV Ju.M.
TITLE On the completeness of the neighborhood spiices Il.
PERIODICAL Trudy Moskovsk. mat. ObgY, - -436 (1955)
reviewed 5/1956 Ij- 421
The author remarks that the aonstrunation of the S-extension cP of theif-space
P given in the first part of this work (Trudy Moskovsk. mat. Obdd..L 271-3o6
(1954)) can be accomplished in an analogous manner by use of the c-ends if
one uses instead of the system Z 0 of all uniform 6-coverings of P an arbitrary
lfpseudo-uniform" structure on P, 1.e. a system E of coverings of P with the
propertiesg Every covering, having an element of 23 inscribed, belongs to E I
to every CKEZ there exists a P62: star-inscribed in CC ; two sets A,BSP
are 6-neighbored exactly then if in every g*E E there exists a re r- with -
r nA o, r eNB / o. Such a E is especially given by every uniform structure
on F (compare Smixnov, Mat.Sbornik, n. Ser, 31, 543-574 (1952)). Analogous to
OF = Z C, P for every E one obtains a 6 -extension 2;P of P. It is the
greatest of all 6-extensions of P on which every open covering of E can be
continued to an open covering. E P is the cf-space of the completion of
the pseudo-uniform space Pr -(P,Z:). From this there result two criteria of
completeness for (pseudo-) uniform spaces. The o-mappings of part I can also
be generalized analogously to M -mappings. There are exactly those mappings
which can be extended to continuous mappings of Z:P in EIPI. Pj:is precompact

Call Nr: AF 1111088125
Transactions of the Third All-union Mathematical Congress ( Cont. ) Moscow,
Jun-Jul '56, Trudy '56, V. 1, Sect. Rpts., Izdatel'stvo ATISSSR, Moscov, 1956, 237 pp.
There are 11 references, all of them USSR.
lAkhtenbaum, L. M. (Moscow). Characteristic Numbers of
Improper Graph.
Smirnov, Yu. M. (Moscow). On the Extension of
_T_o_p-6-lbgi-074Z- Spaces.
Smirnov, Yu. M. (Moscow). on Metrisation of Local
Compact Spaces Which are Decomposable into the Sum of
Countable Mmber of Sets With Countable Bases. 136-137
Mention is made of Aleksandrov, P. S. and Uryson, P. S.
Fet, A. I. (Novosibirsk). Calculus of Variations in the
Large. 137
Mention is made of Lyusternik, L. A., Shnirellman, Shvarts,
A. S., Allber, S. I. and Pontryagin, L. S.
Card 44/80

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SUBJECT USSR/MATHEMATICS/Topology CARD 1/1 PG - 764
AUTHOR SM1IRNOV Ju.M.
TITLE On the metric dimension in the sense of P.S.Alexandro7.
PERIODICAL Izvestija Aked.Nauk 20, 679-684 (1956)
reviewed 5/1957
In the sense of Alexandrov the metric dimension of a given point set M is
the smallest integer r 3;0 for which there exists an arbitrarily small
E-displacement of the set M into a locally finite r-dimensional polyhedron.
The invariant of the set M which is defined in this way, is denoted with
c5mM. The author proves the inequation of Uryson
S m(A UB) _4 ~ mA + 5 mB + I
and the theorem:
For every set A of the metric space M there exists a set HS M of the ty-De GS
such that ASH and JmH = SmA.
The theorem Is an analogue to an earlier result of Tumarkin (Math.Ann. 98,
640-656 (1925))-

SUBJECT USSR/MLTHED1ATICS/TopQ1ogy CARD -1/1 PG - 069
AUTHOR SMIRNOV Ju.M.
TITLE On the dimension of the neighborhood spaces.
PERIODICAL Mat. Sbornik, n. Ser. 16L 283-302 (1956)
reviewed 6/1956
The paper contains in essential the detailed proofs for the most of the
theorems on the ; -dimension SdP of a S-space P, given in an earlier
paper of the author (Doklady Akad. Nauk 22L 717-720(1954)). The most inter-
esting part in most of the proofs is naturally the theorem thatAsdP - dim uP.-
For the theorems on the dimension of a completely-regular space R the author
corrects an error in the earlier paper: The S -structure induced from R
onto a closed set AGR must not be maximal S -structure on A. The theorem
on the dimension R (monotonyq sum theorem) hold also under the stronger
assumption that the subspaces A of R considered in them are completely-closed,
i.e. that every bounded function being defined on A can be continued
continuously on the whole R. For these theorems on the dimension of
completely-regulary spaces compare also Kat9tov (dasopis Mat.Fys, Praha J5L
79-87).

SUBJZCT USSR/MATRRUATICS/Topology CARD 1/1 PG 988
AUTHOR SMIRNOV Yu.M.
TITLE -The~g-e-bo-mi-e-t-ryof infinite uniform complexes and the -dimension
of point sets.
PERIODICAL Mat.Sbornik,n.Ser. -4-0j. 13T-156 (1956)
reviewed 7/1957
In an earlier publication (Dokl:dy Akad.N&uk 25.. 717-720 (1954)) the author
has introduced the notion of th X-dimension for S -spaces and he has
announced some theorems about them. Most of these theorems in the meantime
are proved in detail (of. Mat.Bbornik,n.Ser. 18L 283 (1956))- In the present
paper the author proves the following announced theorem on the ~'-dimenaion of
point sets A in the Euclidean Rns For the fact that A.%R11 has the &-dimension
n it is necessary and sufficient that there exists a number r~o-O such that
for every f- > 0 in the Rn there exists a sphere of the radius r in which
A is an Ewnet. For the proof the so-called uniform complexes T. R11 are
uued. These are such ones for which the simplexes are bounded from above
and below in a oertain manner. The geometric properties of these complexes
are investigated,

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SMItiiiOv, Yuriy Mikhaylovich
Sci for 20 Jun ~`j defense of
pi~vportional topology by the
-Mos State Univ imeni LomonosDv;
(.bimVul 6-583 10)
-- awarded sci degree of Doe Physical-Math
dissertation: "Research in -
_,eno-ral and
integument method" at the CounciL,
Prot No 1) 11 Jan 58.

AUTHOR: SUIRNOV,Yu. 20.6-4/47
TITLE: Example of a Onedimensional Normal Space Which is Contained
in no One-dimensional Bicompactum (Primer odnomernogo normallnogo
prostranstva, ne soderzhashchegosya ni v kakom odnomernom
bikompakte)
PERIODICAL: Doklady Akademii Nauk SSSR,1957,V&-117, Nr 6, PF 439-942 (USSR)
ABSTRACT; The author comprehends onedimensional in the sense of the
inductive dimension according to Uryson (induction with respeot
to points).
Vedenisov IrRef-4] has proved for zerodimensional normal spaces
that they always have a bicompact extension. In the present
paper the author constructs a onedimensional normal space which
is contained in no onedimensional bicompactum. Therewith all
efforts to extend the general result of Wallmen ftef.21 on
normal spaces with the dimension dim defined by coverings to
general normal spaces with the inductive dimension ind can be
omitted.
4 Soviet and 2 foreign references are quoted.
PnSZVTED: By P.S.Aleksandrov, Academician, 21 June 1957
SUBMITTED: 15 June 1957
AVAILABLE: Library of Congress
Card 1/1

AUTHORt Smirnov, Yu SoV/20-120-6-10/59
TITLE: amp e of a Non-Semibicompact Completely Regular Space
'With Zero Dimensional 66khov ccpplement(Primer vpolne regulyar.-
nogo prostranstva s nullmernym chekhovskim narostom, ne
obladayushchego svoystvom. semibikompaktnosti)
PERIODICAL: Doklady Akademii nauk SSSR,1958,Vol 120,Nr 6,pp 1204-1206(USSR)
ABSTRACT: By construction of an example the author shows that the result
of Freudenthal [Ref 1,2] (see the preceding review) is not right
in the,general case.
Starting from the normal null-dimensional space M of Dowker
[Ref 3) with ind M = 0, dim M >0 the author constructs with
the aid of a point A a normal space P with ind (P~A) = 0 but
ind P> 0 (Here ind is understood in the sense of Uryson, it is
defined by induction with respect to points. It always,holds
ind R,