L-paracompactness and L2-paracompactness

A topological space X is called L-paracompact if there exist a paracompact space Y and a bijective function f : X -> Y such that the restriction f (sic)(A) : A -> f(A) is a homeomorphism for each Lindelof subspace A subset of X. A topological space X is called L-2-paracompact if there exist a Hausdorff paracompact space Y and a bijective function f : X -> Y such that the restriction f (sic)(A): A -> f(A) is a homeomorphism for each Lindelof subspace A subset of X. We investigate these two properties.