Abstract Menger's conjecture that Menger spaces are σ-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. We define co-analytic and co-K-analytic spaces. For non-metrizable spaces, analytic Menger spaces are σ-compact, but Menger continuous images of co-analytic spaces need not be. The general co-analytic case is still open, but many special cases are undecidable, in particular, Menger co-analytic topological groups. We also give numerous characterizations of proper K-Lusin spaces, suggesting such spaces are a good answer to the question of “what is a suitable class of ‘definable’ spaces in the non-metrizable context?”. Our methods include the Axiom of Co-analytic Determinacy, non-metrizable Descriptive Set Theory, and Arhangel'skiĭ's work on generalized metric spaces.

中文翻译：

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协解析空间、K 解析空间和 Menger 猜想的可定义版本

摘要 Menger关于Menger空间是σ-紧的猜想是错误的；对于波兰空间的解析子空间是正确的，对于波兰空间的更复杂的可定义子空间是不可判定的。我们定义了协解析和协 K 解析空间。对于不可度量空间，解析Menger空间是σ-紧的，但协解析空间的Menger连续图像不一定是σ-紧的。一般的协解析情况仍然是开放的，但许多特殊情况是不可判定的，特别是门格尔协解析拓扑群。我们还给出了适当 K-Lusin 空间的许多特征，表明这些空间是对“在不可度量化的上下文中什么是合适的‘可定义’空间类？”这个问题的一个很好的回答。我们的方法包括共分析确定性公理、不可度量的描述集理论和 Arhangel'skiĭ'