Metacompactness in Countable Products

Abstract:

      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.