On Lindelöf-normal spaces

Abstract

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 G δ -set or all countable subsets of Y are closed. A compact space X such that X 3 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= ∏ {X n: 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.