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When do connected spaces
have nice connected preimages?


Author: Vladimir V. Tkachuk
Journal: Proc. Amer. Math. Soc. 126 (1998), 3437-3446
MSC (1991): Primary 54A25
DOI: https://doi.org/10.1090/S0002-9939-98-04476-1
MathSciNet review: 1459154
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Abstract: We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly $\sigma $-discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly $\sigma $-discrete connected topology. Another result is that a finer Tychonoff connected strictly $\sigma $-discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space $\mathcal{X}$ is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on $\mathcal{X}$.


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  • [Ar] A.V.Arhangel'skii, Continuous mappings, factorization theorems and function spaces (in Russian), Trudy Mosk. Mat. Obsch., 1984, vol. 47, 3-21. MR 86i:54002
  • [Arh] A.V.Arhangel'skii, Structure and classification of topological spaces and cardinal invariants (in Russian), Uspehi Mat. Nauk, 1978, v.33, No. 6, 29-84. MR 80i:54005
  • [ArPo] A.V.Arhangel'skii, V.I.Ponomarev, Fundamentals of General Topology, Problems and Exercises, Reidel P.C., Dordrecht, 1984. MR 56:3781
  • [CoGa] W.W.Comfort, S.García-Ferreira, Resolvability: a selective survey and some new results, Topology and Its Applications, 1996, vol. 74, 149-167. MR 97i:54005
  • [vDa] E.K. van Douwen, Remote points, Dissertationes Mathematicae, vol. 188, 1-45. MR 83i:54024
  • [Dr] A.N.Dranishnikov, Absolute extensors in dimension $n$ and $n$-soft maps increasing the dimension (in Russian), Uspehi Mat. Nauk, 1984, vol.39, no. 5, 55-95. MR 86c:54017
  • [En] R.Engelking, General Topology, PWN, Warszawa, 1977. MR 58:18316b
  • [GuReSt] J.A.Guthrie, D.F.Reynolds and H.E.Stone, Connected expansions of topologies, Bull. Austral. Math. Soc., 1973, vol. 9, 259-265. MR 48:7201
  • [Ju] H.J.K.Junnila, Stratifiable preimages of topological spaces, Colloq. Math. Soc. J. Bolyai 23. Topology, Budapest, 1978, 689-703. MR 81m:54019
  • [Ko] Kok, H., Connected Orderable Spaces, Math. Centrum Tracts, no. 49, Amsterdam, 1973. MR 49:3862
  • [Pa] B.A.Pasynkov, Zero-dimensional open maps which increase dimension (in Russian), Uspehi Mat. Nauk, 1963, vol.18, no. 5, 183-190. MR 27:6244
  • [PeSh] V.G.Pestov, D.B.Shakmatov, Continuous homomorphic images of second countable groups do not represent all groups with countable network (in Russian), Vestnik Mosk. Univ., Mat., Mech., 1986, no.3, 98-101. MR 87j:22003
  • [STTWW] D.B.Shakhmatov, M.G.Tka\v{c}enko, V.V.Tkachuk, S.Watson, R.G.Wilson, Neither first countable nor \v{C}ech-complete spaces are maximal Tychonoff connected, to appear in Proceedings of the Amer. Math. Soc. CMP 97:11
  • [Us] V.V.Uspensky, For any $X$ the product $X\times Y$ is homogeneous for some $Y$, Proceedings of the Amer. Math. Soc., 1983, vol. 87, no. 1, 187-188. MR 84b:54075
  • [Ve] N.V.Velichko, On the theory of resolvable spaces (in Russian), Mat. Zametki, 1976, vol. 19, 109-114. MR 54:11282
  • [WaWi] S.Watson, R.G.Wilson, Embeddings in connected spaces, Houston J. Math., 1993, vol.19, no.3, 469-481. MR 94k:54040

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Additional Information

Vladimir V. Tkachuk
Affiliation: Departamento de Matematicas, Universidad Autónoma Metropolitana, Av. Michoacan y La Purísima, Iztapalapa, A.P. 55-532, C.P. 09340, Mexico, D.F.
Email: vova@xanum.uam.mx

DOI: https://doi.org/10.1090/S0002-9939-98-04476-1
Keywords: Connected space, metric space, open map, preimage, finer connected topology, homogeneous space
Received by editor(s): November 14, 1996
Received by editor(s) in revised form: April 4, 1997
Communicated by: Alan Dow
Article copyright: © Copyright 1998 American Mathematical Society

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