Add to ORKGAbstract. In this paper, the author discusses spaces with certain ℵ0-weak bases, gives some characterizations of images of metric spaces by certain quotient maps, and studies the covering properties of spaces with special ℵ0-weak bases. The main results are the following: (1) X is a quotient, boundary-countable s-image of a metric space if and only if X is a quo-tient, countable-to-one image of a metric space; (2) A regular space X has a σ-locally finite ℵ0-weak base if and only if X is a quotient, countable-to-one(σ-compact; boundary-σ-compact), σ- image of a metric space; (3) X is a quotient, σ-compact image of a metric space if and only ifX has a point-finite ℵ0-weak development. (4) Every regular space (resp., normal space) with a σ-clo...