We study preservation properties of Namba forcing on κ. We prove that if $\mathcal{I}$ is an ideal with a Borel base on ${\omega }^{\omega }$ and $\kappa >{\omega }_1$ is a regular cardinal less than the uniformity number or bigger than the covering number of $\mathcal{I}$, then the κ-Namba forcing preserves the covering of $\mathcal{I}$. The situation at $\kappa ={\omega }_1$, also treated here, is more complex.

中文翻译：

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难波强迫的守恒定理

我们研究了难波强迫κ的保存特性。我们证明$\mathcal{一世}$ 是一个理想的Borel基地 ${\omega }^{\omega }$ 和 $\kappa >{\omega }_{\mathrm{1个}}$ 是小于统一数或大于覆盖数的常规基数 $\mathcal{一世}$，则κ- Namba强迫保留了$\mathcal{一世}$。情况$\kappa ={\omega }_{\mathrm{1个}}$，也在这里处理，比较复杂。