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Keywords:
extraresolvable; $\kappa$-resolvable
Summary:
Following Malykhin, we say that a space $X$ is {\it 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 {\it 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 $G(\kappa,\kappa)= \{x \in \{0,1\}^\kappa : |\{ \xi < \kappa : x_\xi \neq 0 \}| < \kappa \}$ 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 $\kappa$.
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