We prove that \(C_p(X)\) is pseudocomplete if and only if it has a dense cofinally Polish subspace. This result provides positive answers to two open questions from (Niknejad in Bull Belg Math Soc 25(3):439–452, 2018). We also establish that a space X is cofinally Polish if and only if its Hewitt extension \(\upsilon X\) is cofinally Polish and show that a subspace X of an ordinal is cofinally Polish if and only if X has countable extent.

中文翻译：

---

当且仅当空间$$ C_p（X）$$ C p（X）共抛光时

我们证明\（C_p（X）\）是伪完全的，当且仅当它具有密集的共同最终波兰子空间时。该结果为来自（Niknejad in Bull Belg Math Soc 25（3）：439-452，2018）中的两个悬而未决的问题提供了肯定的答案。我们还建立了空间X当且仅当其休伊特扩展名\（\ upsilon X \）是共同最终抛光时才是共同抛光的，并证明当且仅当X具有可数范围时，序数子空间X才是最终抛光。