Countable compactness of lexicographic products of GO-spaces

Kemoto, Nobuyuki

Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 3, pp. 421-439.

Résumé

We characterize the countable compactness of lexicographic products of GO-spaces. Applying this characterization about lexicographic products, we see: \begin{itemize} \item[$\circ$] the lexicographic product $X^2$ of a countably compact GO-space $X$ need not be countably compact, \item[$\circ$] $\omega_1^2$, $\omega_1\times \omega$, $(\omega+1)\times (\omega_1+1)\times\omega_1\times \omega$, $\omega_1\times \omega\times \omega_1$, $\omega_1\times \omega\times\omega_1\times \omega\times \cdots $, $\omega_1\times \omega^\omega$, $\omega_1\times \omega^\omega\times (\omega+1)$, $\omega_1^\omega$, $\omega_1^\omega\times (\omega_1+1)$ and $\prod_{n\in \omega}\omega_{n+1}$ are countably compact, \item[$\circ$] $\omega\times \omega_1$, $(\omega+1)\times (\omega_1+1)\times\omega\times \omega_1$, $\omega\times \omega_1\times\omega\times \omega_1\times \cdots $, $\omega\times \omega_1^\omega$, $\omega_1\times \omega^\omega\times \omega_1$, $\omega_1^\omega\times \omega$, $\prod_{n\in \omega}\omega_{n}$ and $\prod_{n\leq \omega}\omega_{n+1}$ are not countably compact, \item[$\circ$] $[0,1)_\mathbb R\times \omega_1$, where $[0,1)_\mathbb R$ denotes the half open interval in the real line $\mathbb R$, is not countably compact, \item[$\circ$] $\omega_1\times [0,1)_\mathbb R$ is countably compact, \item[$\circ$] both $\mathbb S\times \omega_1$ and $\omega_1\times \mathbb S$ are not countably compact, \item[$\circ$] $\omega_1\times (-\omega_1)$ is not countably compact, where for a GO-space $X=\langle X,$, $-X$ denotes the GO-space $\langle X,>_X,\tau_X\rangle$. \end{itemize}

Détail

DOI : 10.14712/1213-7243.2019.020

Classification : 54B05, 54B10, 54C05, 54F05
Keywords: lexicographic product; GO-space; LOTS; countably compact product