The Smirnov remainders of uniformly locally connected proper metric spaces

Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line

“…We have just seen thatμ : [1,2] × βN 0 −→ H t is a continuous surjection. The next task is to find out when two points of [1,2] Let us first show thatμ(2, U ) =μ(1, 1 + U ).…”

“…We have just seen thatμ : [1,2] × βN 0 −→ H t is a continuous surjection. The next task is to find out when two points of [1,2] Let us first show thatμ(2, U ) =μ(1, 1 + U ). Take f ∈ U. Theñ Each interval [2 n , 2 n+1 ) contains exactly one point of the form c2 n with n ∈ N 0 and another one of the form d2 n .…”

“…Let us denote by N 0 the set of all nonnegative integers. Every point t ∈ [1, ∞) can be written as t = c · 2 n for some n ∈ N 0 and c ∈ [1,2]. Let m : [1,2] × N 0 −→ [1, ∞) be the map sending (c, n) to c · 2 n and notice that m(c, n) = m(d, k) if and only if c = 2, d = 1 and k = n + 1 or vice-versa.…”

“…Every point t ∈ [1, ∞) can be written as t = c · 2 n for some n ∈ N 0 and c ∈ [1,2]. Let m : [1,2] × N 0 −→ [1, ∞) be the map sending (c, n) to c · 2 n and notice that m(c, n) = m(d, k) if and only if c = 2, d = 1 and k = n + 1 or vice-versa. Composing m with the mapping [1, ∞) −→ H(U) sending t to t −1 δ t we obtain a map µ : [1,2]…”

“…Since the case where U or V are fixed is trivial this leads to the following description of H t . Note that every continuous surjection between Hausdorff compacta is automatically a quotient map (cf.Willard [16, Chapter 3, §9]).⋆ The fiber H t is homeomorphic to the quotient obtained from[1,2] × βN 0 after identifying each point of the form (2, U ) with (…”