26 references, page 1 of 3 with δ = 1, 2 are disjoint. Let us put ti,δ = zi,δ(r) for all i = 1, . . . , k and δ = 1, 2.

Since the set S = {sj : j ∈ ω} is dense in Π, we can choose, for every i ≤ n and ǫ ǫ δ = 1, 2, an element si,δ ∈ S ∩ ti,δW ∩ W ti,δ. Then si,δ ∈ ti,δW for each ǫ = ±1.

Furthermore, according to our choice of the variables zi, the elements si,δ can be chosen to satisfy si,δ = sl,δ whenever di = dl, where i, l ≤ n and δ = 1, 2. It now follows from O1 ∩ O2 = ∅ that the elements (ii) there exist rational numbers 0 = b0 < b1 < · · · < bm−1 < bm = 1 such that f is constant on each subinterval [bk, bk+1) and f (bk) = gk ∈ D for k = 0, 1, . . . , m − 1.

for every r ∈ [0, 1]. Hence µ ({x ∈ J : Φ(r, x) ∈/ U }) < ǫ for each r ∈ [0, 1]. In other words, the path Φ lies in O(U, ε), so the set O(U, ε) is path-connected. Since the sets of the form G0 ∩ O(U, ε) constitute a base for G0 at the identity, this completes the proof of the theorem.

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[3] Wistar W. Comfort and Gerald L. Itzkowitz, Density Character in Topological Groups, Math. Ann. 226 (1977), 223-227.

[4] Richard Engelking, On the double circumference of Alexandroff, Bull. Acad. Pol. Sci. S´er. Math. 16 (1968), 629-634.

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[6] Jorge Galindo,t Mikhail Tkachenko, Montserrat Bruguera, and Constancio Herna´ndez, Extensions of reflexive P -groups, Topol. Appl. 163 (2014), 112-127.