414/514 Homework 1 – The reals

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 (A\mid B) in a cut of \mathbb Q, that is, a “real” is a set A\subset \mathbb Q that is non-empty, bounded above, closed to the left (meaning, if x\in A, y\in\mathbb Q, and y<x, then y\in A), and such that A has no maximum. We also have a copy \mathbb Q^* of \mathbb Q inside \mathbb R, given by the identification q\mapsto q^*:=\{t\in\mathbb Q\mid t<q\}. We left a few details to be verified when we presented this construction.

Let r be a real (in the sense just described). Define carefully the real -r (meaning, describe -r as a specific set of rationals, and verify that it is a real in the sense under discussion), and verify that -r+r=0, and that -r is the only real with this property.

Define carefully the product rs of reals r and s, and verify that the distributive property holds.

Check that \mathbb R is Dedekind-complete, that is, any cut of \mathbb R is realized. (S0, ignoring the formal difference between \mathbb Q and \mathbb Q^*, this version of \mathbb R is the Dedekind-completion of \mathbb Q, 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 \mathbb R(x) of  rational functions admits a completion, call it \hat{\mathbb R}(x). Can we extend the addition operation on \mathbb R(x) so it is defined in all of \hat{\mathbb R}(x) and makes it into an abelian group? Can we extend the order so \hat{\mathbb R}(x) 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 q_0,q_1,\dots and r_0,r_1,\dots are equivalent iff q_0,r_0,q_1,r_1,\dots is a Cauchy sequence.

Verify that this is indeed an equivalence relation, and that, given equivalent sequences \vec q and \vec r, 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 (X,<), in the order topology the open sets are (by definition) those subsets of X that are union of (bounded or unbounded) open intervals in X. Show that a linear order (X,<) is order isomorphic to \mathbb R iff the following three properties are verified:

  • X has no first or last elements.
  • X is connected, that is, we cannot write X=A\cup B where A and B are open, nonempty, and disjoint.
  • X is separable, that is, there is a countable subset of X that is dense in X.

 

 

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