Symmetric g-functions and cardinal inequalities.

In this paper, we prove that the cardinality of a space X with a symmetric g -function such that ∩ { g 2 ( n , x ) : n ∈ ω } = { x } is at most c if X satisfies one of the following conditions: (1) X has countable chain condition; (2) X is star countable (even star σ -compact); (3) X is DCCC (defined below) and normal space. We also prove that if X is a DCCC space with a symmetric g -function such that ∩ { g 3 ( n , x ) : n ∈ ω } = { x } then the cardinality of X is at most c . Finally, we make some observations on Moore spaces. [ABSTRACT FROM AUTHOR]