Predictable network, monotonic monolithicity and D-spaces

Abstract

We investigate different concepts related to D-spaces. We show that any space with an ω0-Noetherian point network is hereditarily thickly covered. To distinguish monotonically monolithic spaces and spaces with predictable network, we show that there exists a monotonically monolithic space which doesn't have a predictable network under the assumption 2ω0<2ω1. This provides a consistent negative answer to one of Guo and Junnila's questions in [16]. Some sufficient conditions for a strongly monotonically monolithic space to have a point-countable base are also given.