An application of the Stone-Weierstrass Theorem
DOI:
https://doi.org/10.22199/issn.0717-6279-5576Keywords:
$G_{\delta}$-closed set, compact-Hausdorff, Stone-Weierstrass theorem, hereditarily Lindel\Abstract
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|ℵ0
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