Abstract Let X be a Tychonoff space. We show that C p ( X ) , the space of real-valued continuous functions on X equipped with the pointwise topology, admits a resolution (an ordered covering indexed by N N ) consisting of convex compact sets that swallows the local null sequences if and only if X is countable and discrete. Then we prove (main result) (i) the weak* bidual of C p ( X ) admits a resolution consisting of bounded sets if and only if X is countable. Hence ( i i ) if C p ( X ) admits a fundamental bounded resolution, X must be countable [8] . If X is first countable, then C p ( X ) admits a resolution made up of bounded sets swallowing the Cauchy sequences if and only if X is countable. In the context of the present research, the weak* bidual of C p ( X ) has received little or null attention so far. Result (i) fixes this situation.

中文翻译：

---

C(X) 和它的二元性中存在很好的分辨率通常意味着 C(X) 的可度量性

摘要 令 X 为 Tychonoff 空间。我们证明了 C p ( X ) 是 X 上配备逐点拓扑的实值连续函数空间，允许分辨率（由 NN 索引的有序覆盖）由凸紧集组成，当且仅当且仅如果 X 是可数且离散的。然后我们证明（主要结果）（i）C p ( X ) 的弱*二元性承认由有界集组成的分解当且仅当 X 是可数的。因此 (ii) 如果 C p ( X ) 承认基本有界分辨率，则 X 必须是可数的 [8] 。如果 X 首先是可数的，那么当且仅当 X 是可数的，那么 C p ( X ) 承认由吞噬柯西序列的有界集组成的分解。在本研究的背景下，到目前为止，C p ( X ) 的弱*双对很少或没有受到关注。结果 (i) 解决了这种情况。