具有指定可分解性质的Tychonoff扩张
Tychonoff Expansions with Prescribed Resolvability Properties
AI总结：
该文利用KID扩张方法，证明任意满足特定条件的Tychonoff空间可构造出五种具有不同可分解性质的Tychonoff扩张，从而否定地回答了两个经典问题。
摘要：
近期文献提供了具体且手工构造的Tychonoff空间（在ZFC中）的例子，这些例子对以下问题给出否定回答，分别归因于Ceder和Pearson（1967年）以及Comfort和García-Ferreira（2001年）：（1）每个$\omega$-可分解空间是否都是最大可分解的？（2）每个最大可分解空间是否都是超可分解的？现在，利用${\mathcal{KID}}$扩张方法，作者证明{\it 每个}适当限制的Tychonoff拓扑空间$(X,\sT)$都有一个更大的Tychonoff拓扑（即一个“扩张”）来见证这种失败。具体地，作者在ZFC中证明：如果$(X,\sT)$是一个最大可分解的Tychonoff空间，且$S(X,\sT)\leq\Delta(X,\sT)=\kappa$，则$(X,\sT)$具有Tychonoff扩张$\sU=\sU_i$（$1\leq i\leq5$），满足$\Delta(X,\sU_i)=\Delta(X,\sT)$和$S(X,\sU_i)\leq\Delta(X,\sU_i)$，使得$(X,\sU_i)$满足：（$i=1$）$\omega$-可分解但不是最大可分解；（$i=2$）[若$\kappa'$是正则的，且$S(X,\sT)\leq\kappa'\leq\kappa$]对所有$\tau<\kappa'$是$\tau$-可分解，但不是$\kappa'$-可分解；（$i=3$）最大可分解但不是超可分解；（$i=4$）超可分解但不是最大可分解；（$i=5$）最大可分解且超可分解，但不是强超可分解。
英文摘要：
The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and García-Ferreira (2001): (1) Is every $ω$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of ${\mathcal{KID}}$ expansion, the authors show that {\it every} suitably restricted Tychonoff topological space $(X,\sT)$ admits a larger Tychonoff topology (that is, an "expansion") witnessing such failure. Specifically the authors show in ZFC that if $(X,\sT)$ is a maximally resolvable Tychonoff space with $S(X,\sT)\leqΔ(X,\sT)=κ$, then $(X,\sT)$ has Tychonoff expansions $\sU=\sU_i$ ($1\leq i\leq5$), with $Δ(X,\sU_i)=Δ(X,\sT)$ and $S(X,\sU_i)\leqΔ(X,\sU_i)$, such that $(X,\sU_i)$ is: ($i=1$) $ω$-resolvable but not maximally resolvable; ($i=2$) [if $κ'$ is regular, with $S(X,\sT)\leqκ'\leqκ$] $τ$-resolvable for all $τ<κ'$, but not $κ'$-resolvable; ($i=3$) maximally resolvable, but not extraresolvable; ($i=4$) extraresolvable, but not maximally resolvable; ($i=5$) maximally resolvable and extraresolvable, but not strongly extraresolvable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
