A REMARK ON TANAKA'S QUESTION(Set-theoretic Topology and Geometric Topology)
Masami Sakai · Institutional Repositories DataBase (IRDB) · 1995
Tanalca posed the following question.Question Is every space with a locally countable $\mathrm{k}$ -network a a-space ?It is known that every $\mathrm{k}$ -space with a locally countable $\mathrm{k}$ -network is the topological sum of $\mathrm{N}_{0^{-\mathrm{s}\mathrm{a}}}\mathrm{p}\mathrm{c}\mathrm{e}\mathrm{S}$ (hence, a a-space), and there is a space with a locally countable k-network which is not an $\aleph$ -space.In this note, we remark that we can find counterexamples for the question under some set theoretic axioms.The author does not know any counterexample in ZFC.For terminology and notions, see the $\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}[2]$ of Tanaka.We have only to find a space X with the following (1) locally countable( $\mathrm{i}.\mathrm{e}$ .every point of X has a countable neighborhood), (2) every compact subset of X is a finite set, (3) not perfect ( $\mathrm{i}.\mathrm{e}$ .there is an $\mathit{0}$ pen subset of X which is not an $F_{\sigma}$ -set).From ( 1) and ( 2), the family $\{\{x\} : x\in X\}$ is a locally countable $\mathrm{k}$ -network of X, and (3) means that X is not a a-space.For cardinals $\alpha,$ $\beta$ , we set $[\alpha]^{\beta}=\{A:A\subset \mathrm{a}, |A|=\beta\}$ .We endow $\omega_{1}$ with the discrete topology.Let $\mathcal{P}=$ { $P_{\alpha}$ : a $<\tau$ } $\subset[\omega_{1}]^{\omega}$ be an almost disjoint family, and choose any $(p_{\alpha})\in$ $\Pi\{P_{\alpha}^{*} : \alpha<\tau\}$ , where $P_{\alpha}^{*}=Cl_{\beta\omega_{1}}P\alpha-P_{\alpha}$ .Then the subspace $X=\omega_{1}\cup\{p_{\alpha} : \alpha<\tau\}$ of $\beta\omega_{1}$ obviously satisfies (1) and ( 2).Moreover, if $(p_{\alpha})$ satisfies the following $(^{*})$ , then X is not perfect.$(^{*})$ For every $A\in[\omega_{1}]^{\omega_{1}}$ , there is an $\alpha<\tau$ such that $A\in p_{\alpha}$ .Thus we have only to find an almost disjoint family $\mathcal{P}=\{P_{\alpha} : \alpha<\tau\}\subset[\omega_{1}]^{\omega}$ and 数理解析研究所講究録