On metrizable type (MT-) maps and spaces

In this paper we define and study MT-maps, which are the fibrewise topological analogue of metrizable spaces, i.e., the extension of metrizability from the category Top to the category TopY. Several characterizations and properties of MT-maps are proved. The notion of an MT-space as an MT-map preimage of a metrizable space is introduced. Examples of MT-spaces and their relation with M-spaces are given. Finally it is deduced that an MT-space with a Gδ-diagonal is metrizable.