A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network. 


Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. 



> 
> Must $X$ be separable?
> 
> 
> 


Thanks very much.



Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. Must $X$ be separable?
If a $\sigma$-space $X$ is Lindelof, then it is paracompact and by Theorem 4.4 of Gruenhage's survey "Generalized Metric Spaces" in the "[Hanbook of Set-Theoretic Topology](http://www.sciencedirect.com/science/book/9780444865809)", $X$ has countable network and hence is hereditarily separable and hereditarily Lindelof.


