On Lindelِf-normal spaces

A space X is Lindelöf-normal or L-normal if every Lindelöf closed subset of X has arbitrarily small closed neighborhoods. It is proved that if the product X×Y is hereditarily L-normal then either every Lindelöf closed subset of X is a regular Gδ-set or all countable subsets of Y are closed. A compact space X such that X3 is hereditarily L-normal is metrizable. By the aid of MA+¬CH it is proved that if exp(X) is hereditarily L-normal then X is a metrizable compact space. A regular space X is called a perfectly L-normal space if the closure of every Lindelöf subset of X is functionally closed. Each perfectly L-normal space is hereditarily L-normal. A product space X=∏{Xn: n∈ω} is perfectly L-normal if and only if all finite subproducts of X are perfectly L-normal. Every hereditarily L-normal dyadic space is metrizable.