Advertisement

Existenc'a de máximo de funciones reales continuas en espacios casi numerablemente compactos

  • 17 Accesses

Summary

By the concept of countable almost-compactness, here defined, a characterization of topological spaces on which each continuous real function has a maximum and minimum is given. Besides a careful study of the relations between several concepts of compactness and axioms of separation is performed.

Bibliografia

  1. [1]

    Alexandroff, A. D.,On the extension of a Hausdorff space to an H-closed space, C.R. (Doklady) « Acad. Sci. U.R.S.S. », N. S. 37 (1942) 118–121.

  2. [2]

    Alexandroff, P. andHopf, H.,Topologie I, Berlin, Springer, 1935.

  3. [3]

    Bourbaki, N.,Topologie générale, Act. Sci. Ind. 858–1142. Paris, Hermann, 1951.

  4. [4]

    Fréchet, M.,Généralisation d'un théorème de Weierstrass, C. R. Acad. Sci. 139 (1904) pág. 848. Este trabajo se encuentra reproducido en la NotaA, pág. 275 de «Les Espaces Abstraites », Paris, Gauthier-Villars, 1928.

  5. [5]

    Hewitt, E.,On two problems of Urysohn, « Ann. of Math. » (2) 47 (1946) 503–509.

  6. [6]

    ——,Ring of real-valued continuous functions I, « Trans. Amer. Math. Soc. » 64 (1948) 45–99.

  7. [7]

    Katetov, M.,Uber H-abgeschlossene und bikompakte Räume, « Časopis Pěst. Mat. Fys. » 69 (1940) 36–49.

  8. [8]

    —— ——,On H-closed extensions of topological spaces, Časopis Pěst. Mat. Fys. 72 (1947) 17–32.

  9. [9]

    Kelley, J. L.,General Topology, New York, D. Van Nostrand Company, Inc., 1955.

  10. [10]

    Novak, J.,Regular space on which every continuous functions is constant, Časopis Pěst. Mat. Fys. 73 (1948) 58–68.

  11. [11]

    Stone, M. H.,The theory of representations for Boolean algebras, « Trans. Amer. Math. Soc, » 40 (1936) 37–111.

  12. [12]

    Tichonoff, A.,Uber die topologische Erweiterung von Räumen, « Math. Ann. », 102 (1929) 544–561.

Apéndice

  1. [1]*

    Aquaro, G.,Funzioni reali continue e spazii topologici pseudocompatti, Atti del VI Congresso dell'Unione Mat. Italiana. Roma, Ed. Cremonese, 1960.

  2. [2]*

    —— ——,Ricovrimenti aperti e strutture uniformi sopra uno spazio topologico, « Annali di Mat. Pura ed Appl. », (IV) Vol. XLII (1959) 319–389.

  3. [3]*

    Bagley, E.H. Connell andJ. D. Mcknight Jr.,On properties characterizing pseudocompact spaces, “ Proc. Amer. Soc. ”, Vol. 9 (1958) 500–506,

  4. [4]*

    Iseki, K. andKasahara, S.,On pseudo-compact and countably compact spaces, « Proc. Japan Acad. », Vol. 33 (1957) 100–102.

  5. [5]*

    Iseki, K.,On weakly compact topological spaces, « Proc. Japan Acad », Vol. 33 (1957) 182.

  6. [6]*

    —— ——,On weakly compact regular spaces I, Proc Japan Acad., Vol. 33 (1957) 252–254.

  7. [7]*

    —— ——,A characterization of pseudo-compact spaces, « Proc. Japan Acad. », Vol. 33 (1957) 320–322.

  8. [8]*

    —— ——,On weakly compact and countably compact topological spaces, « Proc. Japan Acad. », Vol. 33 (1957) 323–324.

  9. [9]*

    Kasahara, S.,Boundedness of semi-continuous finite real functions, « Proc. Japan Acad. », Vol. 33 (1957) 183–186.

  10. [10]*

    —— ——,On weakly compact regular spaces II, « Proc. Japan Acad. », Vol. 33 (1957) 255–259.

  11. [11]*

    Mardesic, S. etPapic, P.,Sur les éspaces dont toute transformation réelle continue est bornée, « Glasnik Matematicko-Fizicki i Astronomski », (II) Vol. 10 (1955) 225–232.

Download references

Author information

Additional information

La redacción de este trabajo nos ha sido simplificada por las notas que tomó nuestra alumna Srta. Callaved de una exposición que hicimos de él en el Seminario Matemático de la Universidad de Zaragoza.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

R.-Salinas, B. Existenc'a de máximo de funciones reales continuas en espacios casi numerablemente compactos. Annali di Matematica 56, 375–413 (1961) doi:10.1007/BF02414282

Download citation