(a) Construct a gauge on 0, 4 that will force the numbers 1, 2, 3 to be tags of any d fine partition of this interval (b) Given a gauge 1 on 0, 4 , construct a gauge 2 such that every 2 fine partition of 0, 4 will (i) have the numbers 1, 2, 3 in its collection of tags, and (ii) be 1 fine ...

(a) Construct a gauge on [0, 4] that will force the numbers 1, 2, 3 to...

Question:

(a) Construct a gauge δ on [0, 4] that will force the numbers 1, 2, 3 to be tags of any d-fine partition of this interval.
(b) Given a gauge δ1 on [0, 4], construct a gauge δ2 such that every δ2-fine partition of [0, 4] will (i) have the numbers 1, 2, 3 in its collection of tags, and (ii) be δ1-fine.