Decide which of the following sets is compact:
(c.i) [1, 2] U {3}, in (R, d), d(x, y) = |x − y|; (c.ii) Qn [0, 1], in (R, d), d(x, y) = |x − y|;
c.iii) {}21 (the closure of the set of all monomials), in (C[0, 1], d), equipped with the uniform metric d(f, g) = maxre[0,1] |f(x) = g(x).
In each case, provide clear arguments and/or counterexample to support your claim.