About uniformly Menger spaces

Precompact type properties - precompactness (=totally precompactness), s-precompactness, pre-Lindelöfness, (=ℵ0-boundedness), t -boundedness - belong to the basic important invariants studied in the uniform topology. The theory of these invariants is wide and continues to develop. However, in a sense, the class of uniformly Menger spaces escaped the attention of researchers. Lj.D.R. Kočinac was the first who introduced and studied the class of uniformly Menger spaces in [3, 4]. It immediately follows from the definition that the class of uniformly Menger spaces lies between the class of precompact uniform spaces and the class of pre-Lindelöf uniform spaces. Therefore, we expect it to have many good properties. In this paper some important properties of the uniformly Menger spaces are investigated. In particular, it is established that under uniformly perfect mappings, the uniformly Menger property is preserved both in the image and the preimage direction.