1. (a) What is the "Continuum Hypothesis"[CH]? [Please state it precisely in your own words.] (b) Now explain why CH implies that if X, Y, and Z are infinite sets of real numbers, then at least two of them match. (c) Assuming the CH and facts about the sets [0, 1] and IRR (the set of irrationsals) proved in class, explain why [0, 1] must match IRR.
2. (a) If < is a binary relation which linearly orders {a, b, c, d, e}, now many ordered pairs must < contain? Explain. (b) If ~ is an equivalence relation on the same set what is the smallest number of ordered pairs it must contain? Explain.
3. Is it possible for a subset X of a linearly ordered pace (S, <) to be both an open interval and a closed interval? Explain. Why or why not?
4. In the "Square Space" ([0, 1]X[0, 1],< x <) is the subset {(x, x): x belongs to [0, 1]} closed? Why or why not?
5. Let H and K denote closed subsets of an arbitrary linearly ordered space (S, <). If X denotes the union of H and K, and Y denotes the intersection of H and K, prove that both of X and Y are necessarily closed sets.
6. Prove or disprove that if f: (S, <) 3 (S, <) is an isomorphism from a linearly ordered space to itself, then f is a continuous function.
7. Prove that if H is a closed interval and K is a closed set disjoint from H. then there exists an open interval, say U, that contains H and is disjoint from K.. [I would prefer a proof for the general case of an arbitrary linearly ordered space (S, <), but I will accept a proof in the special case of the real line with its usual order if easier to explain.]
8. Prove that if p and q and r are distinct points, then there exist disjoint open sets U and V and W such that p is in U, q is in V, and r is in W.
9. Assume that the space ([0, 1], usual order) is connected (which is true). Use this assumption to prove that the "Square Space" is also connected. (Please say as much as you can even if you can't get a complete proof.)
10. Prove (a) that every constant function from any linearly ordered space to itself is continuous and (b) that the "identity" function defined by f(x) = x for every x in the space is also continuous.