Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
connected; locally connected; free topological group; Pontryagin's duality; pseudo-open mapping; open mapping; Urysohn space; connectification
connected; locally connected; free topological group; Pontryagin's duality; pseudo-open mapping; open mapping; Urysohn space; connectification
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.
References:
[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
[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
[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
[4] Bowers P.L.: Dense embeddings of nowhere locally compact metric spaces. Topology Appl. 26 (1987), 1-12. MR 0893799
[5] Emeryk A., Kulpa W.: The Sorgenfrey line has no connected compactification. Comment. Math. Univ. Carolinae 18.3 (1977), 483-487. MR 0461437 | Zbl 0369.54007
[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
[7] Hewitt E., Ross K.A.: Abstract Harmonic Analysis, vol.1. Springer Verlag, Berlin, 1963. Zbl 0416.43001
[8] Hofmann K.H., Morris S.A.: Free compact groups I: Free compact Abelian groups. Topology Appl. 23 (1986), 41-64. MR 0849093 | Zbl 0589.22003
[9] Kuratowski K.: Topology, vol.II. Academic Press, N.Y., 1968. MR 0259835
[10] Malykhin V.I.: On perfect restrictions of mappings (in Russian). Uspekhi Matem. Nauk 40 (1985), 205-206. MR 0783622
[11] Mardešić S.: On covering dimension and inverse limits of compact spaces. Illinois J. Math. 4.2 (1960), 278-291. MR 0116306
[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
[13] Morita K.: On closed mappings and dimension. Proc. Japan Acad. 32 (1956), 161-165. MR 0079755 | Zbl 0071.38501
[14] Morris S.A.: Free Abelian topological groups. in: Proc. Conf. Toledo, Ohio 1983, Heldermann Verlag, Berlin, 1984, pp.375-391. MR 0785024 | Zbl 0802.22001
[15] Okunev O.G.: A method of constructing examples of $M$-equivalent spaces. Topology Appl. 36 (1990), 157-171. MR 1068167
[16] Pontryagin L.S.: Topological Groups. Princeton Univ. Press, Princeton, NY, 1939. Zbl 0882.01025
[17] Porter J., Woods R.G.: Subspaces of connected spaces. to appear. MR 1374077 | Zbl 0855.54025
[18] Robinson D.J.S.: A Course in the Theory of Groups. Graduate Texts in Mathematics vol.80, Springer-Verlag, NY, 1982. MR 0648604 | Zbl 0836.20001
[19] Roy P.: A countable connected Urysohn space with a dispersion point. Duke Math. J. 33 (1966), 331-333. MR 0196701 | Zbl 0147.22804
[20] Tkačenko M.G.: On topologies of free groups. Czechoslovak Math. J. 34 (1984), 541-551. MR 0764436
[21] Watson S., Wilson R.G.: Embedding in connected spaces. Houston J. Math. 19.3 (1993), 469-481. MR 1242433
Search
Partner of
