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# Separation and

Theorem 5.1   The following statements are pairwise equivalent over .
1.
.
2.
SEP3, the third separation theorem.
3.
Let A and B be disjoint, bounded, separably closed, convex sets in R2. Assume also that A is compact. Then A and B can be separated.

Proof. Reasoning in , let A and B satisfy the hypotheses of SEP3. In , separably closed implies closed (see [2]), so B is closed. Hence we can use Heine-Borel compactness of A to find such that for all and . Let be the open ball of radius centered at 0. Then and are disjoint open convex sets. By SEP2 we can strictly separate A' and B'. This proves SEP3 in .

Trivially SEP3 implies statement 5.1.3.

It remains to prove that statement 5.1.3 implies over . Reasoning in , assume that fails. Then there exists a bounded increasing sequence of rational numbers an, , such that does not exist. (See Chapter III of [20].) We may safely assume 0<an<1. Let , and let B be the separably closed convex hull of the points (an,1/n), . Note that A and B are bounded, separably closed, convex sets in R2. Moreover A is compact, and clearly A and B cannot be separated. Thus we have a counterexample to 5.1.3, once we show that A and B are disjoint.

To show that A and B are disjoint, let S be the countable set consisting of all rational convex combinations of points (an,1/n), . Thus B is the separable closure of S.

We claim: for all there exists such that for all , if x<an then . To see this, note that

where , qi>0, . Thus . Putting we have

hence

Furthermore

Therefore we put

and our claim is proved.

Now if , we clearly have x<an for some n. Since S is dense in B, let be such that

Then x'<an+1 and , a contradiction. Thus A and B are disjoint. This completes the proof.

Remark 5.2   A modification of the above argument shows that is equivalent over to the following even weaker-sounding statement: if A and B are disjoint, bounded, separably closed, convex sets in R2, and if A is compact, then there exists an open set U such that and .

Remark 5.3   In the functional analysis literature, separation results such as SEP1, SEP2, and SEP3 are sometimes referred to as geometrical forms of the Hahn-Banach theorem.'' It is therefore of interest to perform a detailed comparison of these separation results with the (non-geometrical) Hahn-Banach and extended Hahn-Banach theorems. Our results in this paper shed some light on the logical or foundational aspect of such a comparison. We note that, although SEP1 and SEP2 are logically equivalent to HB and EHB over (Theorem 4.4), SEP3 is logically stronger (Theorem 5.1). Moreover, even though SEP2 and EHB turn out to be equivalent in this sense, we were unable to find a direct proof of this fact; the proof that we found is highly indirect, via . Thus we conclude that, from our foundational standpoint, it is somewhat inaccurate to view the separation theorems as trivial variants of the Hahn-Banach or extended Hahn-Banach theorems.

Next: Bibliography Up: Separation and Weak König's Previous: Reversal via Hahn-Banach
Stephen G Simpson
1998-10-25