Resolvability in c.c.c. generic extensions

[摘要] Every crowded space $X$ is ${\\omega}$-resolvable in the c.c.c.\\ generic extension $V^{\\operatorname{Fn}(|X|,2)}$ of the ground model. We investigate what we can say about ${\\lambda}$-resolvability in c.c.c.\\ generic extensions for $\\lambda > \\omega$. A topological space is {\\it monotonically $\\omega _1$-resolvable\\/} if there is a function $f:X\\to \\omega _1$ such that \\begin{displaymath} \\{x\\in X: f(x)\\geq {\\alpha}\\}\\subset^{dense}X \\end{displaymath} for each ${\\alpha}< \\omega _1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\\omega}_1$-resolvable in some c.c.c.\\ generic extension; (2) $X$ is monotonically $\\omega _1$-resolvable; (3) $X$ is ${\\omega}_1$-resolvable in the Cohen-generic extension $V^{\\operatorname{Fn}(\\omega _1,2)}$. We investigate which spaces are monotonically $\\omega _1$-resolvable. We show that if a topological space $X$ is c.c.c., and ${\\omega}_1\\le \\Delta(X)\\le |X|<{\\omega}_{\\omega}$, where $\\Delta(X) = \\min\\{ |G| : G \\ne \\emptyset \\mbox{ open}\\}$, then $X$ is monotonically $\\omega _1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=\\Delta(Y)=\\aleph_\\omega$ which is not monotonically $\\omega _1$-resolvable. The characterization of $\\omega _1$-resolvability in c.c.c.\\ generic extension raises the following question: is it true that crowded spaces from the ground model are ${\\omega}$-resolvable in $V^{\\operatorname{Fn}({\\omega},2)}$? We show that (i) if $V=L$ then every crowded c.c.c.\\ space $X$ is ${\\omega}$-resolvable in $V^{\\operatorname{Fn}({\\omega},2)}$, (ii) if there are no weakly inaccessible cardinals, then every crowded space $X$ is ${\\omega}$-resolvable in $V^{\\operatorname{Fn}({\\omega}_1,2)}$. Moreover, it is also consistent, modulo a measurable cardinal, that there is a crowded space $X$ with $|X|=\\Delta(X)=\\omega _1$ such that $X$ remains irresolvable after adding a single Cohen real.

[学科分类] 物理化学和理论化学

[关键词] resolvable;monotonically $\\omega _1$-resolvable;measurable cardinalDOI: DOI 10.14712/1213-7243.2015.226AMS Subject Classification: 54A35 03E35 54A25 PDF