Numbers of arbitrarily large absolute value

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Let K be the Tychonoff-C? ech compactification of R. Show that addition from R × R onto R cannot be extended to a continuous function S from K × K into K . Hint: Let xα be a net in R converging in K to a point x ∈ K \R. Then -xα converges to some point y ∈ K \R. If S exists, then S(x, y) = 0 ∈ R. Then by Problem 4 there must be neighborhoods U of x and V of y in K such that S(u, v) ∈ R and |S(u, v)| <>1 for all u ∈ U and v ∈ V . Show, however, that each of U and V contains real numbers of arbitrarily large absolute value, to get a contradiction.

Reference no: EM13966720

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