#### nomadreid

Gold Member

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[1] Let L be identified with the set of real numbers r, 0 ≤ r ≤ 1, whereby r is in unique decimal form 0.r

_{1}r

_{2}r

_{3}.... , whereby any representation as an infinite sequence 0.s

_{1}s

_{2}......s

_{n}s

_{n+1}00000....., where s

_{n}≠0 & n≥ 1, is excluded, as it is identified with 0.s

_{1}s

_{2}......(s

_{n}-1)99999..... (0 remains 0.000....)

[2] Let each point in S be identified with the ordered pair (a,b), with a, b∈L ,

a = 0.a

_{1}a

_{2}a

_{3}.... , and

b = 0.b

_{1}b

_{2}b

_{3}.... ,

[3] Then the function is (a,b) to c, with c =0.a

_{1}b

_{1}a

_{2}b

_{2}a

_{3}b

_{3}.... ,

that is, if c= 0.c

_{1}c

_{2}c

_{3}.... then for n≥1 , n, c

_{2n-1}=a

_{n}& c

_{2n}=b

_{n}.

(or, to put another way, if a = ∑

_{i=1}

^{∞}a

_{i}×10

^{-i}& b = ∑

_{i=1}

^{∞}b

_{i}×10

^{-i}, then c = ∑

_{i=1}

^{∞}(a

_{i}×10

^{-2i+1}+ b

_{i}×10

^{-2i})

(Corrections in the details would be welcome.)