Recall that a space is:


* "Lindelof", if every open cover has a countable subcover.
* "Linearly Lindelof", if every open cover which is linearly ordered by $\subseteq$ has a countable subcover.
* "weakly Lindelof" if every open cover has a countable subcollection whose union is dense in the space.


Of course every Lindelof space is both weakly Lindelof and Linearly Lindelof. It's easy to see that every space with the *countable chain condition* (i.e., "there are no uncountable families of pairwise disjoint non-empty open sets") is weakly Lindelof and that every space with a Lindelof dense subset is weakly Lindelof. Using this observation one can easily find a weakly Lindelof non-Lindelof space: any separable non-Lindelof space, like the tangent disk plane, or the Cantor tree, will do. These examples are also non-linearly Lindelof, since they contain an uncountable closed discrete subspace.


Finding a Linearly Lindelof non-Lindelof space is harder, and one of the most outstanding problems in set-theoretic topology is whether there exists a normal Linearly Lindelof space which is not Lindelof. A positive answer to this problem would lead to a new construction of a Dowker space, that is a normal space whose product with the unit interval is not normal. 



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> Is there a Linearly Lindelof Tychonoff space which is not weakly Lindelof? 
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All known examples of Linearly Lindelof non-Lindelof spaces seem to be weakly Lindelof. Here are the two most popular ones:


* Mishchenko space is $X=\bigcup\_{n \in \mathbb{N}} (\prod\_{k \leq n} (\omega\_k+1) \times \prod\_{k>n} \omega\_k)$ with the topology inherited from $\prod\_{n \in \mathbb{N}} (\omega\_n+1)$. This is weakly Lindelof because it has a $\sigma$-compact dense subspace.
* (Buzyakova and Gruenhage) Denote with $2^{\aleph\_\omega}$ the product of $\aleph\_\omega$ many copies of the 2-point discrete space. Let $X=\{x \in 2^{\aleph\_\omega}: |x^{-1}(1)| < \aleph\_\omega \}$ with the topology inherited from the usual product topology on $2^{\aleph\_\omega}$. This example is weakly Lindelof for two reasons: first of all, it also has a $\sigma$-compact dense subspace and then it has the countable chain condition.


Most other examples are obtained by tweaking one of those two to get some additional special property, like first-countability (Pavlov), hereditary realcompactness (Arhangel'skii and Buzyakova), etc...



Is there a linearly Lindelof space which is not weakly Lindelof?
(I hope the following comments are still worthwhile @SantiSpadaro)


We say that a space $X$ is dually-$\mathcal{P}$ if for every open neighbourhood assignment $\phi$ of $X$, there is a subspace $Y$ of $X$ with property $\mathcal{P}$ such that $\phi(Y)=X$. The class of dually-$\mathcal{P}$ spaces is denoted by $\mathcal{P}^\*$ and it is named as the dual class to the class $\mathcal{P}$. Also, a class $\mathcal{P}$ of spaces is said to be autodual with respect to neighbourhood assignments if $\mathcal{P}$ coincides with its dual class $\mathcal{P}^\*$.


Thus, a space $X$ is dually $\sigma$-compact if for every open neighbourhood assignment $\phi$ of $X$, there is a $\sigma$-compact subspace $Y$ of $X$ such that $\phi(Y)=X$.


In *Theorem 3.1* of the paper **Alas, O. T., Tkachuk, V. V., & Wilson, R. G. (2006). Covering properties and neighbourhood assignments. In Topology Proc (Vol. 30, No. 1, pp. 25-37)** (<http://topology.nipissingu.ca/tp/reprints/v30/tp30102.pdf>), it was obtained an example of a Tychonoff dually $\sigma$-compact non-Lindelof space (let us refer to this space as $X\_0$ ). On the other hand, in *Proposition 2.7* of the paper **van Mill, J., Tkachuk, V. V., & Wilson, R. G. (2007). Classes defined by stars and neighbourhood assignments. Topology and its Applications, 154(10), 2127-2134** (<https://www.sciencedirect.com/science/article/pii/S0166864107000193>), it was showed that the linearly Lindelof class is autodual. It follows that the space $X\_0$ of Theorem 3.1 above, is a linearly Lindelof space that is not Lindelof. Finally, in *Corollary 2.10* of the paper **Buzyakova, R. Z., Tkachuk, V. V., & Wilson, R. G. (2007). A quest for nice kernels of neighbourhood assignments. Commentationes Mathematicae Universitatis Carolinae, 48(4), 689-697** (<https://dml.cz/bitstream/handle/10338.dmlcz/119691/CommentatMathUnivCarolRetro_48-2007-4_12.pdf>), it was showed that $AD(X\_0)$, the Alexandroff double of $X\_0$, is a dually $\sigma$-compact space that is not weakly Lindelof. Hence, $AD(X\_0)$ is an example of a linearly Lindelof space that is not weakly Lindelof.


