zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The three space problem in topological groups. (English) Zbl 1102.54039
A property $\cal P$ is said to be a shape three space property if, for every topological group $G$ and a closed invariant subgroup $N$ of $G$, the fact that both groups $N$ and $G/N$ have $\cal P$ implies that $G$ also has $\cal P$. It is known that compactness, precompactness pseudocompactness, completeness, connectedness and metrizability are three space properties, and, on the other hand, having a countable network, $\sigma$-compactness, Lindelöfness, countable compactness, sequential compactness, sequential completeness and $\omega$-compactness are not three space properties. In this paper, the authors study some properties of compact, countably compact, pseudocompact and functionally bounded sets which are preserved or destroyed when taking extensions of topological groups. A property $\cal P$ is a three space property for compact (countably compact) sets if all compact (countably compact) subsets of a topological group $G$ have $\cal P$ whenever $G$ contains a closed invariant subgroup $N$ such that the compact (countably compact) subsets of both groups $N$ and $G/N$ have $\cal P$. In Section 3, they show that metrizability is a three space property for compact sets and also a three space property for countably compact sets. In Section 4, they present several examples. For example, they show that under ${\frak p}={\frak c}$, there is an Abelian topological group $G$ and a closed subgroup $N$ of $G$ such that all pseudocompact subspaces of $N$ are finite and the quotient group $G/N$ is countable, but $G$ contains a pseudocompact subspaces of uncountable character.

MSC:
54H11Topological groups (topological aspects)
22A05Structure of general topological groups
54A20Convergence in general topology
54G20Counterexamples (general topology)
WorldCat.org
Full Text: DOI
References:
[1] Arhangel’skii, A. V.: Mappings related to topological groups. Soviet math. Dokl. 9, 1011-1015 (1968)
[2] Arhangel’skii, A. V.: On mappings of dense subspaces of topological products. Dokl. akad. Nauk SSSR 197, 750-753 (1971)
[3] Arhangel’skii, A. V.: On some topological spaces that occur in functional analysis. Russian math. Surveys 31, 14-30 (1976)
[4] Arhangel’skii, A. V.: The structure and classification of topological spaces and cardinal invariants. Russian math. Surveys 33, 33-96 (1978)
[5] Arhangel’skii, A. V.: The frequency spectrum of a topological space and the product operation. Trans. Moscow math. Soc. 40, 163-200 (1981)
[6] Arhangel’skii, A. V.: Classes of topological groups. Russian math. Surveys 36, 151-174 (1981)
[7] Arhangel’skii, A. V.: Continuous mappings, factorization theorems and function spaces. Trudy moskov. Mat. obshch. 47, 3-21 (1984)
[8] Arhangel’skii, A. V.: Function spaces in the topology of pointwise convergence, and compact sets. Uspekhi mat. Nauk 39, 11-50 (1984)
[9] Arhangel’skii, A. V.: R-quotient mappings of spaces with a countable base. Dokl. akad. Nauk SSSR 287, 14-17 (1986)
[10] Benyamini, Y.; Rudin, M. E.; Wage, M.: Continuous images of weakly compact subsets of Banach spaces. Pacific J. Math. 70, 309-324 (1977) · Zbl 0374.46011
[11] Bruguera, M.; Tkachenko, M.: Extensions of topological groups do not respect countable compactness. Questions answers gen. Topology 22, No. 1, 33-37 (2004) · Zbl 1056.54035
[12] Comfort, W. W.; Robertson, L.: Extremal phenomena in certain classes of totally bounded groups. Dissertations math. 272, 1-48 (1988) · Zbl 0703.22002
[13] Dikranjan, D.; Tkachenko, M. G.: Weakly complete free topological groups. Topology appl. 112, 259-287 (2001) · Zbl 0979.54038
[14] Dikranjan, D.; Tkachenko, M. G.; Tkachuk, V. V.: Topological groups with thin generating sets. J. pure appl. Algebra 145, 123-148 (2000) · Zbl 0964.22002
[15] Van Douwen, E.: The integers and topology. Handbook of set-theoretic topology, 111-167 (1984)
[16] Van Douwen, E.: The maximal totally bounded group topology on G and the biggest minimal G-space for abelian groups G. Topology appl. 34, 69-91 (1990) · Zbl 0696.22003
[17] Eda, K.; Ohta, H.; Yamada, K.: Prime subspaces in free topological groups. Topology appl. 62, 163-171 (1995) · Zbl 0821.22001
[18] Engelking, R.: General topology. (1989) · Zbl 0684.54001
[19] Fremlin, D. H.: Consequences of martin’s axiom. (1984) · Zbl 0551.03033
[20] Gerlitz, J.; Juhász, I.: On left-separated compact spaces. Comment. math. Univ. carolin. 19, 53-62 (1978)
[21] Glicksberg, I.: Stone -- čech compactifications of products. Trans. amer. Math. soc. 90, 369-382 (1959) · Zbl 0089.38702
[22] Graev, M. I.: Free topological groups. Amer. math. Soc. transl. Ser. 1 8, 305-364 (1962)
[23] Grant, D. L.: Topological groups which satisfy an open mapping theorem. Pacific J. Math. 68, 411-423 (1977) · Zbl 0375.22002
[24] Guran, I.: On topological groups close to being Lindelöf. Soviet math. Dokl. 23, 173-175 (1981) · Zbl 0478.22002
[25] Hardy, J. P.; Morris, S. A.; Thompson, H. B.: Application of the stone -- čech compactification to free topological groups. Proc. amer. Math. soc. 55, 160-164 (1976) · Zbl 0333.22001
[26] Galindo, J.; Hernández, S.: On a theorem of Van douwen. Extracta math. 13, No. 1, 115-123 (1998) · Zbl 0961.22001
[27] Hewitt, E.; Ross, K. A.: Abstract harmonic analysis I. Grundlehrender math. Wiss. 115 (1963) · Zbl 0115.10603
[28] Hodel, R. E.: Cardinal functions I. Handbook of set-theoretic topology, 1-62 (1984)
[29] Ismail, M.; Nyikos, P.: On spaces in which countably compact sets are closed, and hereditary properties. Topology appl. 11, 281-292 (1980) · Zbl 0434.54018
[30] Kunen, K.: Set theory. (1980) · Zbl 0443.03021
[31] Mack, J.; Morris, S. A.; Ordman, E. T.: Free topological groups and the projective dimension of locally compact abelian group. Proc. amer. Math. soc. 40, 303-308 (1973) · Zbl 0263.22001
[32] Morita, K.: On closed mappings and dimension. Proc. Japan acad. Sci. 32, 161-165 (1956) · Zbl 0071.38501
[33] Morris, S. A.: Pontryagin duality and the structure of locally compact abelian groups. London math. Soc. lecture notes ser. 29 (1977)
[34] P. Nickolas, M. Tkachenko, The character of free topological groups I, Appl. General Topology 6 (1) (2005), in press · Zbl 1077.22001
[35] Okunev, O. G.: Topology appl.. 49, 191-192 (1993)
[36] Pestov, V. G.: Some properties of free topological groups. Moscow univ. Math. bull. 37, 46-49 (1982)
[37] Protasov, I. V.: Discrete subsets of topological groups. Math. notes 55, No. 1 -- 2, 101-102 (1994) · Zbl 0836.22003
[38] Robinson, D. J. F.: A course in the theory of groups. (1982) · Zbl 0483.20001
[39] Roelcke, W.; Dierolf, S.: Uniform structures on topological groups and their quotients. (1981) · Zbl 0489.22001
[40] Shapirovskiĭ, B. E.: On mappings onto tychonoff cubes. Uspekhi mat. Nauk 35, 122-130 (1980)
[41] Simon, P.: A compact Fréchet space whose square is not Fréchet. Comment. math. Univ. carolin. 21, 749-753 (1980) · Zbl 0466.54022
[42] Tkachenko, M. G.: A property of compacta. Seminar on general topology, 149-156 (1981)
[43] Tkachenko, M. G.: Generalization of the comfort -- ross theorem I. Ukr. mat. Zh. 41, 377-382 (1989) · Zbl 0698.22004
[44] Tkachenko, M. G.: Introduction to topological groups. Topology appl. 86, 179-231 (1998)
[45] Uspenskij, V. V.: Extensions of topological groups with a countable net. Moscow univ. Math. bull. 39, 84-85 (1984) · Zbl 0575.22012
[46] Vaughan, J. E.: Small uncountable cardinals and topology. Problems in topology, 195-218 (1990)
[47] Yamada, K.: Metrizable subspaces of free topological groups on metrizable spaces. Topology proc. 23, 379-409 (1998) · Zbl 0970.54032