Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Extending families of disjoint zero sets

Author: C. E. Aull
Journal: Proc. Amer. Math. Soc. 89 (1983), 510-514
MSC: Primary 54C50; Secondary 54C45, 54D60, 54G05, 54G10
MathSciNet review: 715876
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The $ z$-cellularity $ z(X)$ of a space $ X$ is defined as

$\displaystyle z(X) = \sup \left\{ {\left\vert Z \right\vert:Z \subset Z(X)} \right\}$

where $ Z(X)$ is the family of zero sets of $ X$. It is proved using CH that a Tychonoff space $ S$ is $ T{C^ * }$-embedded in every Tychonoff space it is $ C$-embedded in iff $ z(S) \leqslant c$. A space $ S$ is defined to be $ T{C^ * }$-embedded in a space $ X$ if any disjoint family of zero sets of $ S$ can be extended to a family of disjoint zero sets of $ X$. Similar theorems are proved for $ {C^ * }$-embedding when $ S$ is a $ P$-space or the zero sets have the Isiwata property.

References [Enhancements On Off] (What's this?)

  • [1] C. E. Aull, Extendability and expandability (submitted for publication).
  • [2] Robert L. Blair, Spaces in which special sets are 𝑧-embedded, Canad. J. Math. 28 (1976), no. 4, 673–690. MR 0420542,
  • [3] Robert L. Blair and Anthony W. Hager, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41–52. MR 0385793,
  • [4] R. Engelking, Cartesian products and dyadic spaces, Fund. Math. 57 (1965), 287–304. MR 0196692
  • [5] -, General topology, PWN, Warsaw, 1977.
  • [6] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
  • [7] Takesi Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455–480; correction, ibid. 23 (1967), 630–631. MR 0219044
  • [8] I. Juhász, Cardinal functions in topology, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg; Mathematical Centre Tracts, No. 34. MR 0340021
  • [9] R. Pol, Short proofs of two theorems on cardinality of topological spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 1245–1249 (English, with Russian summary). MR 0383333
  • [10] Michael L. Wage, Extremally disconnected 𝑆-spaces, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976), Math. Dept., Auburn Univ., Auburn, Ala., 1977, pp. 181–185. MR 0458392
  • [11] R. Grant Woods, A survey of absolutes of topological spaces, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 116, Math. Centrum, Amsterdam, 1979, pp. 323–362. MR 565852

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54C50, 54C45, 54D60, 54G05, 54G10

Retrieve articles in all journals with MSC: 54C50, 54C45, 54D60, 54G05, 54G10

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society