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 $\kappa$.
@article{119084,
author = {Ofelia Teresa Alas and Salvador Garc\'\i a-Ferreira and Artur Hideyuki Tomita},
title = {Extraresolvability and cardinal arithmetic},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {40},
year = {1999},
pages = {279-292},
zbl = {0976.54004},
mrnumber = {1732649},
language = {en},
url = {http://dml.mathdoc.fr/item/119084}
}
Alas, Ofelia Teresa; García-Ferreira, Salvador; Tomita, Artur Hideyuki. Extraresolvability and cardinal arithmetic. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 279-292. http://gdmltest.u-ga.fr/item/119084/
On maximally resolvable spaces, Fund. Math. 55 (1964), 87-93. (1964) | MR 0163279 | Zbl 0139.40401
Topological Groups, in Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, Eds., North-Holland, Amsterdam, 1984, pp.1143-1263. | MR 0776643 | Zbl 1071.54019
Resolvability: a selective survey and some new results, Topology Appl. 74 (1996), 149-167. (1996) | MR 1425934
Strongly extraresolvable groups and spaces, manuscript submitted for publication, 1998. | MR 1803240
Dense subsets of maximally almost periodic groups, to appear in Proc. Amer. Math. Soc. | MR 1707513
The Theory of Ultrafilters, Grudlehren der Mathematischen Wissenschaften vol.211, Springer-Verlag, Berlin, 1974. | MR 0396267 | Zbl 0298.02004
On the maximal resolvability of products of topological spaces, Soviet Math. Dokl. 10 (1969), 659-662. (1969) | MR 0248726 | Zbl 0199.57302
Extraresolvable spaces, to appear in Topology Appl. | MR 1733807 | Zbl 0941.54003
A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333. (1943) | MR 0008692 | Zbl 0060.39407
Irresolvability is not descriptively good, manuscript submitted for publication.
On resolvability of topological spaces, manuscript submitted for publication. | Zbl 1012.54004
On maximally resolvable spaces, Proc. Steklov Institute of Mathematics 154 (1984), 225-230. (1984) | Zbl 0557.54002