Countable compactness of lexicographic products of GO-spaces

the lexicographic product $X^2$ X^2 of a countably compact GO-space $X$ X need not be countably compact,

${\omega }_1^2$ \omega _1^2 , ${\omega }_1\times \omega$ \omega _1\times \omega , $(\omega +1)\times ({\omega }_1+1)\times {\omega }_1\times \omega$ (\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 , ${\omega }_1\times \omega \times {\omega }_1\times \omega \times \cdots$ \omega _1\times \omega \times \omega _1\times \omega \times \cdots , ${\omega }_1\times {\omega }^{\omega }$ \omega _1\times \omega ^\omega , ${\omega }_1\times {\omega }^{\omega }\times (\omega +1)$ \omega _1\times \omega ^\omega \times (\omega +1) , ${\omega }_1^{\omega }$ \omega _1^\omega , ${\omega }_1^{\omega }\times ({\omega }_1+1)$ \omega _1^\omega \times (\omega _1+1) and ${\prod }_{n\in \omega }{\omega }_{n+1}$ \prod _{n\in \omega }\omega _{n+1} are countably compact,

$\omega \times {\omega }_1$ \omega \times \omega _1 , $(\omega +1)\times ({\omega }_1+1)\times \omega \times {\omega }_1$ (\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\times \omega \times \omega _1\times \cdots , $\omega \times {\omega }_1^{\omega }$ \omega \times \omega _1^\omega , ${\omega }_1\times {\omega }^{\omega }\times {\omega }_1$ \omega _1\times \omega ^\omega \times \omega _1 , ${\omega }_1^{\omega }\times \omega$ \omega _1^\omega \times \omega , ${\prod }_{n\in \omega }{\omega }_n$ \prod _{n\in \omega }\omega _{n} and ${\prod }_{n\le \omega }{\omega }_{n+1}$ \prod _{n\le \omega }\omega _{n+1} are not countably compact,

${[0,1)}_{ℝ}\times {\omega }_1$ [0,1)_\mathbb {R}\times \omega _1 , where ${[0,1)}_{ℝ}$ [0,1)_\mathbb {R} denotes the half open interval in the real line $ℝ$ \mathbb {R} , is not countably compact,

${\omega }_1\times {[0,1)}_{ℝ}$ \omega _1\times [0,1)_\mathbb {R} is countably compact,

both $𝕊\times {\omega }_1$ \mathbb {S}\times \omega _1 and ${\omega }_1\times 𝕊$ \omega _1\times \mathbb {S} are not countably compact,

${\omega }_1\times (-{\omega }_1)$ \omega _1\times (-\omega _1) is not countably compact, where for a GO-space $X=\langle X,{<}_X,{\tau }_X\rangle$ X=\langle X,<_X,\tau _X\rangle , $-X$ -X denotes the GO-space $\langle X,{>}_X,{\tau }_X\rangle$ \langle X,>_X,\tau _X\rangle .