Abstract:Following Malykhin, we say that a space $X$ is {extraresolvable} if $X$ contains a family $\Cal D$ of dense subsets such that $|\Cal D| > \Delta (X)$ and the intersection of every two elements of $\Cal D$ is nowhere dense, where $\Delta (X) = \min \{|U|: U$ is a nonempty open subset of $X\}$ is the {dispersion character} of $X$. We show that, for every cardinal $\kappa $, there is a compact extraresolvable space of size and dispersion character $2^\kappa $. In connection with some cardinal inequalities, we prove the equivalence of the following statements: \newline 1) $2^\kappa < 2^{{\kappa }^{+}}$, 2) $(\kappa ^{+})^{\kappa }$ is extraresolvable and 3) $A(\kappa ^{+})^{\kappa }$ is extraresolvable, where $A(\kappa ^{+})$ is the one-point compactification of the discrete space $\kappa ^{+}$. For a regular cardinal $\kappa \geq \omega $, we show that the following are equivalent: 1) $2^{< \kappa } < 2^{\kappa }$; 2) $G(\kappa ,\kappa )$ is extraresolvable; 3) $G(\kappa ,\kappa )^\lambda $ is extraresolvable for all $\lambda < \kappa $; and 4) there exists a space $X$ such that $X^\lambda $ is extraresolvable, for all $\lambda < \kappa $, and $X^\kappa $ is not extraresolvable, where $G(\kappa ,\kappa )= \{x \in \{0,1\}^\kappa : |\{ \xi < \kappa : x_\xi \not =0 \}| < \kappa \}$ for every $\kappa \geq \omega $. It is also shown that if $X$ is extraresolvable and $\Delta (X) = |X|$, then all powers of $X$ have a dense extraresolvable subset, and $\lambda ^\kappa $ contains a dense extraresolvable subspace for every cardinal $\lambda \geq 2$ and for every infinite cardinal $\kappa $. For an infinite cardinal $\kappa $, if $2^\kappa > {\frak c}$, then there is a totally bounded, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa $, and if $\kappa = \kappa ^\omega $, then there is an $\omega $-bounded, normal, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa $.
Keywords: extraresolvable, $\kappa $-resolvable
AMS Subject Classification: Primary 54A35, 03E35; Secondary 54A25