A question on spaces with a point countable base?

Question

Is there a normal Hausdorff space with a point countable base and a dense Lindelöf subspace which is not second countable?

Thanks for your help.

Answer 1

It seems the following.

There exists a (regular) Lindelöf space $X$ with a point countable base which is not second countable (see [GMT, Ex. 9.4] or [Sak, Ex. 2.3]). Let $P$ be a Bernstein set of the unit interval $I=[0,1]$. In other words, $P$ is an uncountable set which contains no uncountable closed set of $I$. Let $X$ be the space obtained from $I$ by isolating the points of $P$. Obviously $X$ has a point-countable base. Note that every open set of $X$ containing $X\setminus P$ is co-countable, hence $X$ is Lindelöf.

Moreover, Eric K. van Douwen, Franklin D. Tall and William A. R. Weiss in [DTW] under CH constructed a nonmetrizable hereditarily Lindelöf space with a point-countable base.

References

[DTW] Eric K. van Douwen, Franklin D. Tall, William A. R. Weiss. Nonmetrizable hereditarily Lindelöf spaces with point-countable bases from CH. Proc. Amer. Math. Soc., 64:1 (May 1977), 139–145.

[GMT] G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984), 303–332.

[Sak] Masami Sakai. Counterexamples on generalized metric spaces. Scientiae Mathematicae Japonicae Online, e-2006, 539–542.