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Title: Connectedness and local connectedness of topological groups and extensions (English)
Author: Alas, O. T.
Author: Tkačenko, M. G.
Author: Tkachuk, V. V.
Author: Wilson, R. G.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 40
Issue: 4
Year: 1999
Pages: 735-753
Category: math
Summary: It is shown that both the free topological group $F(X)$ and the free Abelian topological group $A(X)$ on a connected locally connected space $X$ are locally connected. For the Graev's modification of the groups $F(X)$ and $A(X)$, the corresponding result is more symmetric: the groups $F\Gamma(X)$ and $A\Gamma(X)$ are connected and locally connected if $X$ is. However, the free (Abelian) totally bounded group $FTB(X)$ (resp., $ATB(X)$) is not locally connected no matter how ``good'' a space $X$ is. The above results imply that every non-trivial continuous homomorphism of $A(X)$ to the additive group of reals, with $X$ connected and locally connected, is open. We also prove that any dense in itself subspace of the Sorgenfrey line has a Urysohn connectification. If $D$ is a dense subset of $\{0,1\}^{\frak c}$ of power less than $\frak c$, then $D$ has a Urysohn connectification of the same cardinality as $D$. We also strengthen a result of [1] for second countable Tychonoff spaces without open compact subspaces proving that it is possible to find a compact metrizable connectification of such a space preserving its dimension if it is positive. (English)
Keyword: connected
Keyword: locally connected
Keyword: free topological group
Keyword: Pontryagin's duality
Keyword: pseudo-open mapping
Keyword: open mapping
Keyword: Urysohn space
Keyword: connectification
MSC: 22A05
MSC: 54C10
MSC: 54C25
MSC: 54D06
MSC: 54D25
MSC: 54H11
idZBL: Zbl 1010.54043
idMR: MR1756549
Date available: 2009-01-08T18:57:06Z
Last updated: 2012-04-30
Stable URL:
Reference: [1] Alas O.T., Tkačenko M.G., Tkachuk V.V., Wilson R.G.: Connectifying some spaces.Topology Appl. 71.3 (1996), 203-215. MR 1397942
Reference: [2] Arhangel'skiĭ A.V.: Mappings and spaces (in Russian).Uspekhi Matem. Nauk 21 (1966), 133-184. English translat.: Russian Math. Surveys 21, 115-162. MR 0227950
Reference: [3] Arhangel'skiĭ A.V.: On relations between invariants of topological groups and their subspaces (in Russian).Uspekhi Matem. Nauk 35.3 (1980), 3-22. English translat.: Russian Math. Surveys 35, 1-23. MR 0580615
Reference: [4] Bowers P.L.: Dense embeddings of nowhere locally compact metric spaces.Topology Appl. 26 (1987), 1-12. MR 0893799
Reference: [5] Emeryk A., Kulpa W.: The Sorgenfrey line has no connected compactification.Comment. Math. Univ. Carolinae 18.3 (1977), 483-487. Zbl 0369.54007, MR 0461437
Reference: [6] Graev M.I.: Free topological groups (in Russian).Izvestiya Akad. Nauk SSSR, Ser. Matem. 12 (1948), 279-324. English translat.: Amer. Math. Soc. Transl. (1) 8 (1962), 305-364. MR 0025474
Reference: [7] Hewitt E., Ross K.A.: Abstract Harmonic Analysis, vol.1.Springer Verlag, Berlin, 1963. Zbl 0416.43001
Reference: [8] Hofmann K.H., Morris S.A.: Free compact groups I: Free compact Abelian groups.Topology Appl. 23 (1986), 41-64. Zbl 0589.22003, MR 0849093
Reference: [9] Kuratowski K.: Topology, vol.II.Academic Press, N.Y., 1968. MR 0259835
Reference: [10] Malykhin V.I.: On perfect restrictions of mappings (in Russian).Uspekhi Matem. Nauk 40 (1985), 205-206. MR 0783622
Reference: [11] Mardešić S.: On covering dimension and inverse limits of compact spaces.Illinois J. Math. 4.2 (1960), 278-291. MR 0116306
Reference: [12] Markov A.A.: On free topological groups (in Russian).Izvestiya Akad. Nauk SSSR, Ser. Matem. 9 (1945), 1-64. English translat.: Amer. Math. Soc. Transl. (1) 8 (1962), 195-272. MR 0025474
Reference: [13] Morita K.: On closed mappings and dimension.Proc. Japan Acad. 32 (1956), 161-165. Zbl 0071.38501, MR 0079755
Reference: [14] Morris S.A.: Free Abelian topological Proc. Conf. Toledo, Ohio 1983, Heldermann Verlag, Berlin, 1984, pp.375-391. Zbl 0802.22001, MR 0785024
Reference: [15] Okunev O.G.: A method of constructing examples of $M$-equivalent spaces.Topology Appl. 36 (1990), 157-171. MR 1068167
Reference: [16] Pontryagin L.S.: Topological Groups.Princeton Univ. Press, Princeton, NY, 1939. Zbl 0882.01025
Reference: [17] Porter J., Woods R.G.: Subspaces of connected appear. Zbl 0855.54025, MR 1374077
Reference: [18] Robinson D.J.S.: A Course in the Theory of Groups.Graduate Texts in Mathematics vol.80, Springer-Verlag, NY, 1982. Zbl 0836.20001, MR 0648604
Reference: [19] Roy P.: A countable connected Urysohn space with a dispersion point.Duke Math. J. 33 (1966), 331-333. Zbl 0147.22804, MR 0196701
Reference: [20] Tkačenko M.G.: On topologies of free groups.Czechoslovak Math. J. 34 (1984), 541-551. MR 0764436
Reference: [21] Watson S., Wilson R.G.: Embedding in connected spaces.Houston J. Math. 19.3 (1993), 469-481. MR 1242433


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