Metacompactness in Countable Products

DOI：10.3770/j.issn:2095-2651.2015.06.011

 作者 单位 王建军 四川农业大学数学系, 四川 雅安 625014

主要研究亚紧空间的可数乘积性, 首先证明如果$Y$是遗传亚紧空间且$\{X_n:n\in\omega\}$是由\v{C}ech-散射亚紧构成的可数空间族, 则一下结论等价:(1) $Y\times\prod_{n\in\omega}X_n$ 是亚紧的,(2) $Y\times\prod_{n\in\omega}X_n$ 是可数亚紧的,(3) $Y\times\prod_{n\in\omega}X_n$ 是 ortho-紧的.进而推广了文献 [Tanaka, Tsukuba. J. Math., 1993, 17: 565--587] 中的主要结果. 最后, 证明如果$Y$是遗传$\sigma$-亚紧空间且$\{X_n:n\in\omega\}$是由\v{C}ech-散射$\sigma$-亚紧空构成的可数空间族, 则乘积$Y\times\prod_{n\in\omega}X_n$是$\sigma$-亚紧的.

In this paper, we present that if $Y$ is a hereditarily metacompact space and $\{X_n:n\in\omega\}$ is a countable collection of \v{C}ech-scattered metacompact spaces, then the followings are equivalent: (1)~~$Y\times\prod_{n\in\omega}X_n$ is metacompact, (2)~~$Y\times\prod_{n\in\omega}X_n$ is countable metacompact, (3)~~$Y\times\prod_{n\in\omega}X_n$ is orthocompact. Thereby, this result generalizes Theorem 5.4 in [Tanaka, Tsukuba. J. Math., 1993, 17: 565--587]. In addition, we obtain that if $Y$ is a hereditarily $\sigma$-metacompact space and $\{X_n:n\in\omega\}$ is a countable collection of \v{C}ech-scattered $\sigma$-metacompact spaces, then the product $Y\times\prod_{n\in\omega}X_n$ is $\sigma$-metacompact.