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Sep 24, 2013 at 19:14 comment added Benjamin Steinberg Of course the primitive idempotents are in bijection with $\mathbb R$ and you can recover f in S by multiplying it against the primitive idempotents. So one has to show that there is no way to take R with the discrete topology into a tychonoff space such that S are the functions that extend ctsly.
Sep 24, 2013 at 18:55 history answered Benjamin Steinberg CC BY-SA 3.0