This set is due in two weeks, on Friday September 26, at the beginning of lecture.
1. Recall that in the construction of the reals via Dedekind cuts, a real is simply the left set of a pair in a cut of , that is, a “real” is a set that is non-empty, bounded above, closed to the left (meaning, if , , and , then ), and such that has no maximum. We also have a copy of inside , given by the identification . We left a few details to be verified when we presented this construction.
Let be a real (in the sense just described). Define carefully the real (meaning, describe as a specific set of rationals, and verify that it is a real in the sense under discussion), and verify that , and that is the only real with this property.
Define carefully the product of reals and , and verify that the distributive property holds.
Check that is Dedekind-complete, that is, any cut of is realized. (S0, ignoring the formal difference between and , this version of is the Dedekind-completion of , and this gives us that it is also the Dedekind completion of itself. )
2. More generally, define the Dedekind completion of a dense order, and verify its existence and uniqueness (up to isomorphism). In particular, the field of rational functions admits a completion, call it . Can we extend the addition operation on so it is defined in all of and makes it into an abelian group? Can we extend the order so is in fact an ordered group? What, if any, is the problem trying to extend multiplication?
3. Recall that in the construction of the reals via Cauchy sequences, a real is an equivalence class of Cauchy sequences of rationals, under the equivalence relation that states that two Cauchy sequences and are equivalent iff is a Cauchy sequence.
Verify that this is indeed an equivalence relation, and that, given equivalent sequences and , the resulting interleaving sequence is equivalent to both. Verify that the (pointwise) definitions of addition and multiplication make sense, and that the resulting set equipped with these operations is indeed a field. Define carefully the ordering relation, and prove that it gives us a field ordering. Finally, verify that the resulting ordered field is indeed Dedekind complete.
4. Recall the construction of the reals described in Street’s paper An efficient construction of real numbers. His short note makes many claims that are not proved there. Provide as many of the missing details as possible.
5. Given a linear order , in the order topology the open sets are (by definition) those subsets of that are union of (bounded or unbounded) open intervals in . Show that a linear order is order isomorphic to iff the following three properties are verified:
- has no first or last elements.
- is connected, that is, we cannot write where and are open, nonempty, and disjoint.
- is separable, that is, there is a countable subset of that is dense in .