As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only if every Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).
As it turns out uniform spaces and more generally Cauchy spaces are always preregular so the Hausdorff condition in these cases reduces to the T0 condition These are also the spaces in which completeness makes sense and Hausdorffness is a natural companion to completeness in these cases Specifically a space is complete if and only if every Cauchy net has at least one limit while a space is Hausdorff if and only if every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place)