On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient s–images of locally separable metric spaces, or k–spaces with a star–countable k–network. The main result is that the following conditions are equivalent: (1) b = ω ₁; (2) t(S _ω × S _ω1 ) > ω; (3) For any pair (X, Y ), which are k–spaces with a point–countable k–network consisting of cosmic subspaces, t(X × Y ) ≤ ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k–space property of products of spaces with certain k–networks could be deduced from the above theorem. Supported by the National Science Foundation of China (No. 10271026)