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**Additivity of metrizability and related properties.**
*(English)*
Zbl 0991.54032

Summary: A topological property \({\mathcal P}\) is called \(n\)-additive in \(n\)th power (or weakly \(n\)-additive) if a topological space \(X\) has \({\mathcal P}\) as soon as \(X^n= \bigcup\{Y_i: i\in n\}\) where all \(Y_i\) have \({\mathcal P}\). If \({\mathcal P}\) is \(n\)-additive in \(n\)th power for all natural \(n\geq 1\), we say that \({\mathcal P}\) is weakly finitely additive.

The main question we deal with in this paper is whether metrizability is weakly finitely additive. It was proved by V. V. Tkachuk [Trans. Am. Math. Soc. 341, No. 2, 585-601 (1994; Zbl 0802.54002)] that it is so in the class of regular spaces with Souslin property. Metrizabilty was also proved by Tkachuk [loc. cit.] to be weakly finitely additive in the class of Hausdorff compact spaces. We generalize this last result, showing that metrizability is weakly finitely additive in the class of regular pseudocompact spaces. We also prove that if \(X^n\) is a regular Lindelöf space then it is metrizable if represented as a union of its \(n\) metrizable subspaces. We show that there is an example of a Tikhonov nonmetrizable space \(X\) such that \(X^n\) is a union of two metrizable subspaces for all \(n\geq 1\). The method of constructing this example can be used to solve several problems stated by Tkachuk [loc. cit.].

The main question we deal with in this paper is whether metrizability is weakly finitely additive. It was proved by V. V. Tkachuk [Trans. Am. Math. Soc. 341, No. 2, 585-601 (1994; Zbl 0802.54002)] that it is so in the class of regular spaces with Souslin property. Metrizabilty was also proved by Tkachuk [loc. cit.] to be weakly finitely additive in the class of Hausdorff compact spaces. We generalize this last result, showing that metrizability is weakly finitely additive in the class of regular pseudocompact spaces. We also prove that if \(X^n\) is a regular Lindelöf space then it is metrizable if represented as a union of its \(n\) metrizable subspaces. We show that there is an example of a Tikhonov nonmetrizable space \(X\) such that \(X^n\) is a union of two metrizable subspaces for all \(n\geq 1\). The method of constructing this example can be used to solve several problems stated by Tkachuk [loc. cit.].

### MSC:

54E35 | Metric spaces, metrizability |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |

54B05 | Subspaces in general topology |

54B10 | Product spaces in general topology |

### Citations:

Zbl 0802.54002
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\textit{Z. Balogh} et al., Topology Appl. 84, No. 1--3, 91--103 (1998; Zbl 0991.54032)

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### References:

[1] | Tkachuk, V. V., Finite and countable additivity of topological properties in nice spaces, Trans. Amer. Math. Soc., 341, 2, 585-601 (1994) · Zbl 0802.54002 |

[2] | Uspenskiǐ, V. V., Pseudocompact spaces with a σ-point-finite base are metrizable, Comment. Math. Univ. Carolin., 25, 2, 261-264 (1984) · Zbl 0574.54021 |

[3] | Shelah, S., A weak version of ♢ which follows from \(2^{ℵ0} < 2^{ℵ1}\), Israel J. Math., 29, 239-247 (1978) · Zbl 0403.03040 |

[4] | Shelah, S., Whitehead groups may not be free, even assuming CH I, Israel J. Math., 28, 193-204 (1977) · Zbl 0369.02035 |

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