It is shown that if $X$ is a first-countable countably compact subspace of ordinals then $C_p\left(X\right)$ is Lindelöf. This result is used to construct an example of a countably compact space $X$ such that the extent of $C_p\left(X\right)$ is less than the Lindelöf number of $C_p\left(X\right)$. This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the $G_{\delta }$ topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf...

Given two topologies, $T_1$ and $T_2$, on the same set X, the intersection topologywith respect to $T_1$ and $T_2$ is the topology with basis $U_1\cap U_2:U_1\in T_1,U_2\in T_2$. Equivalently, T is the join of $T_1$ and $T_2$ in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ${\omega }_1$-compactness in this class of topologies. We demonstrate that the majority of his results generalise...

Ram’ırez-Páramo proved that under GCH for the class of compact Hausdorff spaces i-weight reflects all cardinals [A reflection theorem for i-weight, Topology Proc. 28 (2004), no. 1, 277–281]. We show that in ZFC i-weight reflects all cardinals for the class of compact LOTS. We define local i-weight, then calculate i-weight of locally compact LOTS and paracompact spaces in terms of the extent of the space and the i-weight of open sets or the local i-weight. For locally compact LOTS we find a necessary...