A cardinal inequality for topological spaces involving closed discrete sets

Abstract

Abstract.Let X be a Tx topological space. Let a(X) = sup{a: X has a closed discrete subspace of cardinality a} and v(X) = min{a: A^ can be written as the intersection of a open subsets of X x X); here Ay denotes the diagonal {(x,x): x e X] oi X. It is proved that \X\ < exp^X") v(X)). If, in addition, X is Hausdorff, then X has no more than expia^) viX)) compact subsets.1. Introduction. There are several known relationships among the cardinal functions on a topological space that involve the cardinalities of closed discrete subsets of the space. Among these are Jones's lemma (see, for example, [10, p. 100]) and the recent results of Burke and Hodel [2]. It is the purpose of this note to add to this list of relationships.The authors wish to thank the referee for several useful suggestions and, in particular, for pointing out that Theorem 2.6 is a consequence of Theorem 2.1.Information about cardinal functions on topological spaces appears in Juhasz [6]; we shall use the notation and terminology of this text. We shall henceforth assume that all hypothesized topological spaces are Tx. Any additional separation axioms used in the proof of a theorem will be set forth explicitly in the statement of the theorem. The cardinality of a set X is denoted by | A'|; [A'] will denote the set of two-element subsets of X. Cardinal numbers are identified with the set of ordinals preceding them. The smallest cardinal greater than the cardinal X is denoted by X+. We shall use the following settheoretic theorem, due to Erdös and Rado; see [4].

Spaces with property $(DC(\omega_1))$

“…Diagonal property is useful in estimating the cardinality of a space. For example, Ginsburg and Woods in [6] proved that the cardinality of a space with countable extent and a G δ -diagonal is at most c. Therefore, if X is Lindelöf and has a G δ -diagonal then |X| ≤ c. However, the cardinality of a regular space with the countable Souslin number and a G δ -diagonal need not have an upper bound [11], [12]. Buzyakova in [4] proved that if a space X with the countable Souslin number has a regular G δ -diagonal then the cardinality of X does not exceed c. Recently, Xuan and Shi in [13] show that if X is a DCCC space with a rank 3-diagonal then the cardinality of X is at most c.…”

A note on star Lindelöf, first countable and normal spaces

“…It is well known that the cardinality of a space which has countable extent and a G δ -diagonal is at most c (see [4]). Therefore, a positive answer to Question 1.1 would imply a trivial proof of the above result.…”

Cardinalities of DCCC normal spaces with\newline a rank 2-diagonal

“…For example, Ginsburg and Woods in [4] proved that the cardinality of a space with countable extent and a G δ -diagonal is at most c. Therefore, if X is Lindelöf and has a G δ -diagonal, then the cardinality of X is at most c. However, the cardinality of a regular space with the countable Souslin number and a G δ -diagonal need not have an upper bound (see [7], [8]). Buzyakova in [2] proved that if a space X with the countable Souslin number has a regular G δ -diagonal, then the cardinality of X does not exceed c. Rank 3-diagonal is one type of diagonal property.…”