Compact hausdorff space, Let K be a compact Hausdorff space

Compact hausdorff space

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Let K be a compact Hausdorff space and suppose for some k there are k continuous functions f1,..., fk from K into R such that x i→ ( f1(x ),..., fk (x )) is one-to-one from K into Rk . Let F be the smallest algebra of functions containing f1, f2,..., fk and 1. (a) Show that F is dense in C (K ) for dsup. (b) Let K := S1 := {(cos θ, sin θ ): 0 ≤ θ ≤ 2π } be the unit circle in R2 with relative topology. Part (a) applies easily for k = 2. Show that it does not apply for k = 1: there is no 1-1 continuous function f from S1 into R. Hint: Apply the intermediate value theorem, Problem 14(d) of Section 2.2. For θ consider the intervals [0, π ] and [π, 2π ].

Reference no: EM13966693

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