亚紧空间的可数乘积

In this paper,we present that if Y is a hereditarily metacompact space and{Xn:n∈ω}is a countable collection of Cech-scattered metacompact spaces,then the followings are∏equivalent:(1)Y×∏n∈ωXn is metacompact,(2)Y×∏n∈ωXn is countable metacompact,(3)Y×n∈ωXn is orthocompact.Thereby,this result generalizes Theorem 5.4 inTanaka,Tsukuba.J.Math.,1993,17:565–587].In addition,we obtain that if Y is a hereditarilyσ-metacompact space and{Xn:n∈ω∏}is a countable collection of Cech-scatteredσ-metacompact spaces,then the product Y×n∈ωXn isσ-metacompact.

Metacompactness in Countable Products

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.