I think this is quite awkward, but I decided to ask for help anyway.
This is the problem:
Show that every perfectly normal Lindelöf space has the Souslin property.
I've tried to slove this problem by using the fact that every open subset of a perfectly normal space is also Lindelöf, so the two open neighborhoods which seperates two closed subsets from each other (normality) have countable open cover. Now in worst case all of the open sets which these two countable open cover are obtained from, are disjoint, which is still countable and the wanted property holds. But I know there's something wrong with this and I don't know what to do.
Any help would be greatly appreciated.