Richard N. Ball, Anthony W. Hager
Network character and tightness of the compact--open topology

Comment.Math.Univ.Carolin. 47,3 (2006) 473-482.

Abstract:For Tychonof\text {}f $X$ and $\alpha $ an infinite cardinal, let $\alpha def X := $ the minimum number of $\alpha $ cozero-sets of the \v Cech-Stone compactification which intersect to $X$ (generalizing $\Bbb R$-defect), and let $rt X := \min _\alpha \max (\alpha , \alpha def X)$. Give $C(X)$ the compact-open topology. It is shown that $\tau C(X)\leq n\chi C(X) \leq rtX=\max (L(X),L(X) def X)$, where: $\tau $ is tightness; $n\chi $ is the network character; $L(X)$ is the Lindel\"{o}f number. For example, it follows that, for $X$ \v Cech-complete, $\tau C(X)=L(X)$. The (apparently new) cardinal functions $n\chi C$ and $rt$ are compared with several others.

Keywords: compact-open topology, network character, tightness, defect, Lindel\"of number
AMS Subject Classification: 54C35, 46E10, 22A99, 54D20, 54H11