Skip to main content
SpringerLink
Log in

On properties of infimal topology of a map space

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We study properties of the infimal topology τ inf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τ inf to have the Fréchet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τ μ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τ inf to satisfy the first axiom of countability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birkhoff, G. Lattice Theory (Nauka, Moscow, 1984) [Russian translation].

    MATH  Google Scholar 

  2. Timokhovich, V. L., Frolova, D. S. “On Some Topologies on a Set ofMappings”, Vestn. Beloruss. Gos. Univ., Ser. 1, Fiz. Mat. Inform., No. 3, 84–89 (2009) [in Russian].

    MathSciNet  MATH  Google Scholar 

  3. Kukrak, G. O., Timokhovich, V. L. “Some Topological Properties of a Mapping Space”, Vestn. Beloruss. Gos. Univ., Ser. 1, Fiz.Mat. Inform., No. 1, 144–149 (2010) [in Russian].

    MathSciNet  MATH  Google Scholar 

  4. Frolova, D. S. “On the Supremum Topology of a Mapping Space”, Vestn. Beloruss. Gos. Univ., Ser. 1, Fiz. Mat. Inform., No. 1, 116–117 (2011) [in Russian].

    MathSciNet  MATH  Google Scholar 

  5. Timokhovich, V. L., Frolova, D. S. “On the Infimum Topology of a Mapping Space”, Vestn. Beloruss. Gos. Univ., Ser. 1, Fiz.Mat. Inform., No. 2, 136–140 (2011) [in Russian].

    MathSciNet  MATH  Google Scholar 

  6. Timokhovich, V. L., Frolova, D. S. “Property and Admissibility in Arens–Dugundji’s Sense of the Infimum of a Family of Uniform Convergence Topologies”, Dokl.NANBelarusi 56, No. 2, 22–26 (2012) [in Russian].

    Google Scholar 

  7. Timokhovich, V. L., Frolova, D. S. “Topologies ofUniform Convergence. The Property in the Sense of Arens-Dugundji and the Sequential Property”, Russian Mathematics (Iz. VUZ) 57, No 9, 37–48 (2013).

    MathSciNet  MATH  Google Scholar 

  8. Engelking, R. General Topology (Mir, Moscow, 1986) [Russian translation].

    MATH  Google Scholar 

  9. Hart, K. P., Nagata, J., Vaughan, J. E. Encyclopedia of General Topology (Elsevier, 2004).

    MATH  Google Scholar 

  10. Hausdorff, F. “Erweiterung einer Homöomorphie”, Fund. Math. 16, 353–360 (1930).

    MATH  Google Scholar 

  11. Aleksandrov, P. S., Urysohn, P. S. Memoirs on Compact Topological Spaces (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  12. Arens, R., Dugundji, J. “Topologies for Function Spaces”, Pacif. J.Math. 1, No. 1, 5–31 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  13. Staum, R. “The Algebra of Bounded Continuous Functions Into a Nonarchimedean Field”, Pacif. J. Math. 50, No. 1, 169–185 (1974).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Belarusian State University, pr. Nezavisimosti 4, Minsk, 220030, Republic of Belarus

    V. L. Timokhovich

  2. Business Analyst, EPAM Systems, ul. Akadem. Kuprevicha 1-1, Minsk, 220141, Republic of Belarus

    D. S. Frolova

Authors
  1. V. L. Timokhovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. S. Frolova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. S. Frolova.

Additional information

Original Russian Text © V.L. Timokhovich, D.S. Frolova, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 4, pp. 87–99.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Timokhovich, V.L., Frolova, D.S. On properties of infimal topology of a map space. Russ Math. 60, 72–82 (2016). https://doi.org/10.3103/S1066369X16040113

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X16040113

Keywords

Search

Navigation