Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $SCH$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.

中文翻译：

---

几乎强紧致性与强紧致性还有多远

Bagaria和Magidor引入了几乎强紧性的概念，这与强紧性的概念非常接近。博尼和布鲁克-泰勒询问最不几乎强紧基数是否是强紧基数。戈德堡对此案给出了肯定的答复$SCH$从下往上成立，并且最不几乎强紧紧基数具有不可数共尾性。在本文中，我们对一般情况给出否定的答案。我们的结果也对 Bagaria 和 Magidor 的问题给出了肯定的答案。