论文标题
$ \ ell _ {\ infty} $的某些子集中的密集线索和间隙
Dense lineability and spaceability in certain subsets of $\ell_{\infty}$
论文作者
论文摘要
我们研究了$ \ ell_ \ infty $的子集的密集线索和间隔性，并具有规定的累积点。我们证明，所有有界序列的集合完全可以计，许多积累点在$ \ ell_ \ infty $中都可以固化，从而补充了Papathanasiou的最新结果，Papathanasiou的最新结果证明了对于连续体的序列，许多积累点都相同。我们还证明这些集合是可间隔的。然后，我们考虑具有有限数量的累积点的一组有界非构造序列的问题。我们证明，这种集合在$ \ ell_ \ infty $中密度可统一，但是它仍然不可间隔。在理想收敛的设置和空间$ \ mathbb {r}^ω$中，也研究了上述问题。
We investigate dense lineability and spaceability of subsets of $\ell_\infty$ with a prescribed number of accumulation points. We prove that the set of all bounded sequences with exactly countably many accumulation points is densely lineable in $\ell_\infty$, thus complementing a recent result of Papathanasiou who proved the same for the sequences with continuum many accumulation points. We also prove that these sets are spaceable. We then consider the same problems for the set of bounded non-convergent sequences with a finite number of accumulation points. We prove that such set is densely lineable in $\ell_\infty$ and that it is nevertheless not spaceable. The said problems are also studied in the setting of ideal convergence and in the space $\mathbb{R}^ω$.
