As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family $$\mathcal{U}$$ U of non-empty open subsets of X there is a compact subspace $$K \subset X$$ K ⊂ X such that $$K \cap U \ne \emptyset$$ K ∩ U ≠ ∅ for each $$U \in \mathcal{U}$$ U ∈ U . In this note we answer several questions raised in [7] by showing that (1) any first countable cellular-compact T 2 -space is T 3 , and so its cardinality is at most $$\mathfrak{c} = 2^{\omega}$$ c = 2 ω ; (2) cov $$(\mathcal{M}) > {\omega}_1$$ ( M ) > ω 1 implies that every first countable and separable cellular-compact T 2 -space is compact; (3) if there is no S -space then any cellular-compact T 3 -space of countable spread is compact; (4) $$MA_{\omega{1}}$$ M A ω 1 implies that every point of a compact T 2 -space of countable spread has a disjoint local $$\pi$$ π -base.

中文翻译：

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在蜂窝紧凑空间

正如 Tkachuk 和 Wilson 在 [7] 中介绍的那样，如果对于 X 的非空开放子集的任何细胞，即不相交的族 $$\mathcal{U}$$ U 存在，拓扑空间 X 是细胞紧致的一个紧凑子空间 $$K \subset X$$ K ⊂ X 使得 $$K \cap U \ne \emptyset$$ K ∩ U ≠ ∅ 对于每个 $$U \in \mathcal{U}$$ U ∈ U . 在这篇笔记中，我们回答了 [7] 中提出的几个问题，证明 (1) 任何第一个可数的蜂窝紧凑 T 2 空间是 T 3 ，因此它的基数至多为 $$\mathfrak{c} = 2^{ \omega}$$ c = 2 ω ; (2) cov $$(\mathcal{M}) > {\omega}_1$$ ( M ) > ω 1 意味着每一个第一个可数和可分离的蜂窝紧凑T 2 -空间都是紧的；(3) 如果没有 S -空间，那么任何可数展开的元胞紧致 T 3 -空间都是紧致的；