Given a section f for the projection p from this double-cover to Heath’s V-space, we write V0 , resp. V1 , for the points (x, 0) on the x-axis of Heath’s V-space that the section f carries to the lower, resp. upper, copy in the double cover.
And we write V0, n , resp. V1, n , for the set of points (x, 0) of V0 , resp. of V1 , for which the section f carries the entire basic V neighborhood of (x, 0) , of finger-height 1/n , to the basic broken V neighborhood, of same finger-height, of f(x, 0) in the double cover.
Anyway, the x-axis of Heath’s V-space is the join of V0 with V1 .
And, if f is continuous, each Vi is the join of the (nested) family {Vi, n}n (i = 0, 1) .
Or, at any rate, the set of x-axis points (x, 0) of Vi at which f is continuous is just that join.

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