The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is in ZFC a regular (or just Hausdorff) SP space of cardinality $>\mathfrak{c}$. While this question is left open, we establish a number of non-trivial results that we list below.

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It is consistent with $MA+\mathfrak{c}={\mathrm{\aleph }}_2$ that there is a countably tight and compact SP space of cardinality $2^{\mathfrak{c}}$.

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If κ is a measurable cardinal then in the forcing extension obtained by adding κ many Cohen reals, every countably tight regular SP space has cardinality at most $\mathfrak{c}$.

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If $\kappa >{\mathrm{\aleph }}_1$ Cohen reals are added to a model of GCH then in the extension every pseudocompact SP space with a countable dense set of isolated points has cardinality at most $\mathfrak{c}$.

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If $\mathfrak{c}\le {\mathrm{\aleph }}_2$ then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinality greater than $\mathfrak{c}$.

本文的目的是考虑有关各种可分离伪径向（简称：SP）空间的可能最大基数的问题。这里最有趣的问题是在 ZFC 中是否有一个常规的（或只是 Hausdorff）SP 基数空间$>\mathfrak{C}$. 虽然这个问题是悬而未决的，但我们建立了以下列出的一些重要结果。

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它符合 $米\mathrm{一种}+\mathfrak{C}={\mathrm{\aleph }}_2$存在可数紧且紧的基数SP空间$2^{\mathfrak{C}}$.

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如果κ是可测基数，那么在通过将κ添加许多 Cohen 实数获得的强制扩展中，每个可数紧的正则 SP 空间至多具有基数$\mathfrak{C}$.

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如果 $\kappa >{\mathrm{\aleph }}_1$ 将 Cohen 实数添加到 GCH 模型中，然后在扩展中，每个具有可数稠密孤立点集的伪紧 SP 空间至多具有基数 $\mathfrak{C}$.

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如果 $\mathfrak{C}\le {\mathrm{\aleph }}_2$ 那么有一个 0 维 SP 空间，其中有一个可数的密集孤立点集，其基数大于 $\mathfrak{C}$.