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 ∩{g2(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 ∩{g3(n,x):n∈ω}={x} then the cardinality of X is at most c. Finally, we make some observations on Moore spaces.