An application of the Stone-Weierstrass Theorem


  • Adalberto García-Máynez Universidad Nacional Autónoma de México.
  • Margarita Gary Universidad del Atlántico.
  • Adolfo Pimienta Universidad Simón Bolívar.



$G_{\delta}$-closed set, compact-Hausdorff, Stone-Weierstrass theorem, hereditarily Lindel\


Let (X, τ) be a topological space, we will denote by |X|,|X|K, |X|τ and |X|δ, the cardinalities of X; the family of compacts in X; the family of closed in X, and the family of Gδ-closed in X, respectively. The purpose of this work is to establish relationships between these four numbers and conditions under which two of them coincide or one of them is ≤ c, where c denotes, as usual, the cardinality of the set of real numbers R. We will use the Stone-Weierstrass theorem to prove that: Let (X, τ) be a compact Hausdorff topological space, then |X|δ ≤ |X|0

Author Biographies

Adalberto García-Máynez, Universidad Nacional Autónoma de México.

Instituto de Matemáticas.

Margarita Gary, Universidad del Atlántico.

Facultad de Ciencias Básicas.

Adolfo Pimienta, Universidad Simón Bolívar.

Facultad de Ciencias Básicas y Biomédicas.


A. V. Arhangel’skii, “On the cardinality of bicompacta satisfying the first axiom of countability”, Doklady Akademii Nauk SSSR, vol. 187, 1969.

A. V. Arhangel’skii, “On hereditary properties”, Topology and its Applications, vol. 3, pp. 39-46, 1973.

C. Borges, “On stratifiable spaces”, Pacific Journal of Mathematics, vol. 17, pp. 1-16, 1966.

D. K. Burke and R. E. Hodel, “The number of compact subsets of a topological space”, Proceedings of the American Mathematical Society, vol. 58, pp. 363-368, 1976.

R. Engelking, “Cartesian product and dyadic space”, Fundamenta Mathematicae, vol. 59, pp. 287-304, 1966.

R. Engelking, “On functions defined of cartesian product”, Fundamenta Mathematicae, vol. 59, pp. 221-231, 1966.

F. B. Jones, “On the first countability axiom for locally compact Hausdorff spaces”, Colloquium Mathematicum, vol. 7, pp. 33-34, 1959.

J. Kelley, General Topology. New York: Springer-Verlag, 1975.

K. Kuratowski, Topology, vol. 1. New York: Academic Press, 1966.

J. Prolla, “Weierstrass-stone theorems for set-valued mappings”, Journal of Approximation Theory, vol. 36, pp. 1-15, 1982.

P. Roy, “The cardinality of first countable spaces”, Bulletin of The American Mathematical Society, vol. 77, no 6, pp. 1057-1059, 1971.

E. Schenkman, “The Weierstrass appoximation theorem”, The American Mathematical Monthly, vol. 79, 1972.

M. H. Stone, “Applications of the theory of Boolean rings to general topology”, Transactions of the American Mathematical Society, vol. 41, pp. 375-481, 1937.

M. H. Stone, “The generalized weierstrass approximation theorem”, Mathematics Magazin, vol. 21, no. 5, pp. 237-254, 1948.

K. Weierstrass, Mathematische werke von Karl Weierstrass, vol. 7. Berlín: Mayer and Muller, 1885.

S. Willard, General topology. Addison-Wesley, 1970.



How to Cite

A. García-Máynez, M. Gary, and A. Pimienta, “An application of the Stone-Weierstrass Theorem”, Proyecciones (Antofagasta, On line), vol. 42, no. 5, pp. 1211-1220, Sep. 2023.




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