Given a uniform space in the entourage sense, define a cover to be uniform if there is some entourage such that for each there is an such that These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of as ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other.
Given a uniform space in the entourage sense define a cover to be uniform if there is some entourage such that for each there is an such that These uniform covers form a uniform space as in the second definition Conversely given a uniform space in the uniform cover sense the supersets of as ranges over the uniform covers are the entourages for a uniform space as in the first definition Moreover these two transformations are inverses of each other