Continuous isomorphisms onto separable groups

Luis Felipe Morales López

doi:10.4995/agt.2012.1625

Continuous isomorphisms onto separable groups

Luis Felipe Morales López

DOI: https://doi.org/10.4995/agt.2012.1625

Abstract

A condensation is a one-to-one continuous function onto. We give sufficient conditions for a Tychonoff space to admit a condensation onto a separable dense subspace of the Tychonoff cube Ic and discuss the differences that arise when we deal with topological groups, where condensation is understood as a continuous isomorphism. We also show that every Abelian group G with |G| 2c admits a separable, precompact, Hausdorff group topology, where c = 2!.

Keywords

Condensation; Continuous isomorphism; Separable groups; Subtopology

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References

A. V. Arhangel’skii and M .G. Tkachenko, Topological Groups and Related Structures, Atl. Stud. in Math. 1. (Atlantis Press, Paris, 2008).

A. V. Arhangel’skii, Cardinal Invariants of Topological Groups. Embeddings and Condensations, (Russian), Dokl. Akad. Nauk SSSR 247 (1979), 779–782.

A. V. Arhangel’skii, General Topology III: Paracompactness, Function Spaces, Descriptive Theory, Ency. of Math. Sci., 51, (Springer, Berlin 1995).

A. V. Arhangel’skii, On Countably Compact Topologies on Compact Groups an on Dyadic Compacta, Topology Appl. 57 (1994), 163–181. http://dx.doi.org/10.1016/0166-8641(94)90048-5

D. N. Dikranja and D. B. Shakhmatov, Hewitt-Marczewski-Pondiczery Type Theorem for Abelian Groups and Markov’s Potential Density, Proc. Amer. Math. Soc. 138 (2010), 2979–2990. http://dx.doi.org/10.1090/S0002-9939-10-10302-5

D. N. Dikranjan and D. B. Shakhmatov, Forcing Hereditarily Separable Compact-like Group Topologies on Abelian Groups, Topology Appl. 151 (2005), 2–54. http://dx.doi.org/10.1016/j.topol.2004.07.012

R. Engelking, General Topology, Sigma Ser. Pure Math. 6 (Heldermann, Berlin, 1989).

C. Hernández, Condensations of Tychonoff Universal Topological Algebras, Comment. Math. Univ. Carolin. 42, no. 3 (2001), 529–533.

I. Druzhinina, Condensations onto Connected Metrizable Spaces, Houston J. Math. 30, no. 3 (2004), 751–766.

L. Fuchs, Infinite Abelian Groups, Vol. I, Pure and AppliedMath., Vol. 36-I. (Academic Press, New York-London, 1970).

G. Gruenhage, V. V. Tkachuk and R. G. Wilson, Weaker Connected and Weaker Nowhere Locally Compact Topologies for Metrizable and Similar Spaces, Topology Appl. 120 (2002), 365–384. http://dx.doi.org/10.1016/S0166-8641(01)00082-7

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I, Structure of Topological Groups, Integration Theory, Group Representations. Second edition. Fund. Prin. Of Math. Sci., 115. (Springer-Verlag, Berlin-New York, 1979).

T. Isiwata, Compact and Realcompact -metrizable Extensions. Proc. of the 1985 topology conf. (Tallahassee, Fla., 1985). Topology Proc. 10, no. 1 (1985), 95–102.

V. G. Pestov. An Example of Nonmetrizable Minimal Topological Group whose Identity has the Type G. (Russian) Ukrain. Mat. Zh. 37, no. 6 (1985), 795–796.

D. B. Shakhmatov, Condensations of Universal Topological Algebras Preserving Continuity of Operations and Decreasing Weights, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2), (1984), 42–45.

M. G. Tkachenko, V. V. Tkachuk, V. V. Uspenskij and R. G. Wilson, In Quest of Weaker Connected Topologies, Comment. Math. Univ. Carolin. 37, no. 4 (1996), 825–841.

L. Yengulalp, Coarser Connected Metrizable Topologies, Topology Appl. 157 (14) (2010), 2172–2179. http://dx.doi.org/10.1016/j.topol.2010.06.002

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