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Is there a weakly Lindel\"of Tychonoff Moore non-ccc space?

Note that here ccc denotes the countable chain condition; a space $X$ is called weakly Linde\"of if for any open cover $\mathcal U$ of $X$ there is a counable subset $\mathcal V \subset \mathcal U$ such that $\bigcup \mathcal V$ is dense in $X$.

I have a regular example in hand; but i'm looking for a Tychononff example of a weakly Lindel\"of non-ccc space?

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  • $\begingroup$ Could you describe your "regular" example? Maybe it is Tychonoff automatically? Because existing examples of regular non-Tychonoff spaces are not Moore. $\endgroup$
    – Taras Banakh
    Commented May 11, 2017 at 5:19
  • $\begingroup$ Let $X$ be a regular first countable Lindel\"of non-ccc space. Then the space $M(X)$ is the regular example, where $M(X)$ is the Moore Machine( see xueshu.baidu.com/…) $\endgroup$
    – Paul
    Commented May 11, 2017 at 6:31
  • $\begingroup$ It is a good question under which conditions the Moore Machine prserves the Tychonoff property of spaces. $\endgroup$
    – Taras Banakh
    Commented May 11, 2017 at 7:39

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