| Title:
|
Countable compactness of lexicographic products of GO-spaces (English) |
| Author:
|
Kemoto, Nobuyuki |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
421-439 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $X=\langle X,<_X,\tau_X\rangle$, $-X$ denotes the GO-space $\langle X,>_X,\tau_X\rangle$. \end{itemize} (English) |
| Keyword:
|
lexicographic product |
| Keyword:
|
GO-space |
| Keyword:
|
LOTS |
| Keyword:
|
countably compact product |
| MSC:
|
54B05 |
| MSC:
|
54B10 |
| MSC:
|
54C05 |
| MSC:
|
54F05 |
| idZBL:
|
Zbl 07144904 |
| idMR:
|
MR4034442 |
| DOI:
|
10.14712/1213-7243.2019.020 |
| . |
| Date available:
|
2019-10-29T13:04:08Z |
| Last updated:
|
2021-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147853 |
| . |
| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| . |
