On Lindelöf property and spread in C p-theory

Abstract

The following facts are established. If X is ω-monolithic and w-stable, and the spread of C p ( X) is countable, then X has a countable network. If C p ( X) is a Lindelöf Σ-space and the spread of C p ( X) is countable, then X has a countable network. Under (MA + ┐CH), if s( X) ⩽ ω and C p ( X) is a Lindelöf Σ-space, then X has a countable network. If G is a topological group such that C p ( G) is a Lindelöf Σ-space, then G has a countable network. Under (MA + ┐CH), if X is a Lindelöf p-space of countable tightness, and Y is a subspace of X of the countable spread, then Y is hereditarily Lindelöf and hereditarily separable. Several open problems are formulated.