Why don't you try the same idea as for the Alexandrov compactification? There one considers a locally compact space, one adds an extra point, and one defines the new neighborhoods of the extra point as the complements of the compact subsets of the space (together with the extra point). It is an easy exercize to show that the new space is compact.

So you must define local Lindelöf spaces, where any point belongs to an open set possesing a Lindelöf closure. We add an extra point and we define its neighborhoods as complements of all closed Lindelöf subsets. At the first sight it looks to work, but maybe one has to add more conditions (both to premises and conclusion) in order to make the statement interesting and/or useful.

Good idea... thanks mihai we can make analogousbto construction of compactification for lindelofization. I think it can work even generalised case..

It seems to me that the question of lindelofization is useless since every topological space X is compactifiable i.e there is a compact topological space Y in which X is dense and therefoe Y is a lindelof space.

I do not understand what does generalized spaces mean, but there is a notion of Lindeloffication for point-free spaces or locales and such things can be proved to exist only for non-spatial locales, for spatial locales (which are none else than usual topologies of topological spaces) they do not exist.

If you are working in the Hausdorff context, a space is compactifiable iff it is completely regular.  Mihai's suggestion for taking a "one-point Lindelofication" seems attractive, but what do you do with discrete spaces X?  While you can use the cofinite subsets of X to add to your new point p (to make X\cup {p}), you could also use the cocountable ones. (Of course then X won't be dense in X\cup {p} unless X is uncountable.)  Then you get a Lindelof topology on X\cup {p} that is strictly finer than the Alexandroff topology.  Generally the theory of Lindelof (Hausdorff) spaces is much more unruly than that for compact ones; e.g., Lindelof spaces needn't even be regular.  Stimulating question, though...

It would be interesting to consider maximal lindelof extension also.