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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
R^{2}. 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
HeineBorel 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
a_{n},
,
such that
does not exist. (See
Chapter III of [20].) We may safely assume 0<a_{n}<1. Let
,
and let B be the separably closed convex
hull of the points (a_{n},1/n), .
Note that A and B are
bounded, separably closed, convex sets in
R^{2}. 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
(a_{n},1/n), .
Thus B is the separable closure of S.
We claim: for all
there exists
such that
for all
,
if x<a_{n} then
.
To see
this, note that
where
,
q_{i}>0,
.
Thus
.
Putting
we have
hence
Furthermore
Therefore we put
and our claim is proved.
Now if
,
we clearly have x<a_{n} for some n.
Since S is dense in B, let
be such that
Then
x'<a_{n+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 weakersounding
statement: if
A and
B are disjoint, bounded, separably closed,
convex sets in
R^{2}, 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 HahnBanach theorem.'' It is therefore
of interest to perform a detailed comparison of these separation
results with the (nongeometrical) HahnBanach and extended
HahnBanach 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 HahnBanach or
extended HahnBanach theorems.
Next: Bibliography
Up: Separation and Weak König's
Previous: Reversal via HahnBanach
Stephen G Simpson
19981025