An L-space is a regular hereditarily Lindelof space which is not hereditarily separable. Consistent examples of L-spaces are relatively easy to come by (for example, Suslin Lines), but the first construction of an L-space in ZFC was only given in 2006 by J.T. Moore.

Moore's space has lots of convergent sequences. Actually, it's even Frechet-Urysohn, meaning that every point in the closure of a set can be approximated by a countable convergent sequence lying inside that set (see Theorem 7.8 from  <cite authors="Justin Tatch Moore" mrnumber="2220104" cite="_J. Amer. Math. Soc._ **19** (2006), no. 3, 717--736 (electronic)">_Justin Tatch Moore_, MR 2220104 [**A solution to the $L$ space problem**](http://dx.doi.org/10.1090/S0894-0347-05-00517-5), _J. Amer. Math. Soc._ **19** (2006), no. 3, 717--736 (electronic).</cite>).

However, there are consistent examples of L-spaces which lack any non-trivial convergent sequences and in a pretty extreme way. Take a *Sierpinski subset of the real line* (that is, an uncountable subset of $\mathbb{R}$ having countable intersection with every Lebesgue null set) and provide it with the topology inherited from the *density topology* (that is the topology on $\mathbb{R}$ whose open sets are the measurable sets with Lebesgue density 1 at each one of their points). Then the resulting space is regular and hereditarily Lindelof and has the property that every countable subset is closed discrete (This was first observed in <cite authors="H. E. White, Jr." mrnumber="341379" cite="_Proc. Amer. Math. Soc._ **44** (1974), 454--462">_H. E. White, Jr._, MR 341379 [**Topological spaces in which Blumberg’s theorem holds**](http://www.ams.org/mathscinet-getitem?mr=341379), _Proc. Amer. Math. Soc._ **44** (1974), 454--462.</cite> and <cite authors="Franklin D. Tall" mrnumber="419709" cite="_Pacific J. Math._ **62** (1976), no. 1, 275--284">_Franklin D. Tall_, MR 419709 [**The density topology**](http://www.ams.org/mathscinet-getitem?mr=419709), _Pacific J. Math._ **62** (1976), no. 1, 275--284.</cite>). 

The existence of Sierpinski set is consistent with and independent from ZFC (CH easily provides a Sierpinski set via induction on $\omega_1$ but $MA_{\omega_1}$ kills all Sierpinski sets), so it's natural to ask:

>>Is there in ZFC an L-space where every countable set is closed and discrete?

>>Is there at least a ZFC example of an L-space without any convergent sequences?