From now on we shall treat only biextensional Chu spaces. Most of
what we say here could be applied to general Chu spaces provided we
substitute ``a'' for ``the'' and
(same representation) for =.
However this complicates the story without any real gain. It would be
like studying preordered sets instead of partially ordered sets, where
the notion of *the* least upper bound of a subset must be generalized
to that of a clique of least upper bounds, with no essential advantage
to the elementary theory.

Up to this point we have dealt with *A*, *X*, and
as
unstructured sets, but with the promise that by representing the
elements of *A* as words we would somehow impose structure on *A*.

In the next section we shall define a general but extremely austere notion of structure. The austerity takes some getting used to, so by way of motivation we first consider in this section a less general but more familiar notion of structure which we shall show is preserved by continuous functions. We illustrate this with three examples, pointed sets, posets, and semilattices.

The class of all relational structures of a given signature is closed under both arbitrary product and substructure. Hence if we equip the unstructured set with a relational structure, chosen completely arbitrarily, then this structure is inherited by , and thence by any subset thereof.

For example if we order the set
so that ,
then this
order is inherited by every Chu space over
in the usual way:
just when for all ,
.
This is just the
ordinary inclusion order on 2^{X} when understood as the power set of *X*.

What we shall show is that continuous functions preserve the structure
inherited from .
What is remarkable about this is that *
the notion of continuity was defined without reference to that
structure,* regardless of how rich or combinatorially complex that
structure might be.

Let us begin with the simple case of declaring one element of ,
say 0, to be a constant (definable as a unary relation holding just at
0), making
a pointed set. Pointed sets transform like
ordinary sets except for the requirement that the constant in the
source be mapped to the constant in the target. Now consider
for any set *X*. The induced constant in
is the
constant word .
This is as in universal algebra, where the
*n*-th direct power of an algebra with a constant contains that
constant as a constant *n*-tuple.

Now suppose
appears in ,
call it .
Then every
projection of
onto
sends
to 0. Let
be continuous. Then every projection of
to
must send
to 0, or we would have a projection of
whose composition with
*f* fails to be a projection of .
Hence
must itself
be the constantly zero word.

But this behavior of *f* must hold even if we do not equip
with any
structure, because the definition of continuous function is independent
of any structure we might assign to .
That is, constant words must
be preserved by continuous functions: not only must
always be
mapped to
(of the appropriate length) but
must be
mapped to
and so on.

We now pass from
as a structure with one or more constants to the
above example
ordered by .
Any Chu space (*A*,*r*,*X*) over
this alphabet, viewed as a subset
of the power set of
*X*, becomes a set of subsets of *X*, with the inherited order being
the ordinary inclusion order.

**Proof:** Suppose to the contrary that
in
but the
continuous function
is such that
.
Hence
for some projection
,
but
.
But
must be a projection of ,
whose
just-observed behavior contradicts .

For our third and last example, furnish
with the usual join or
disjunction operation
.
This induces a partial join
operation on
defined as the join (bitwise OR) of words
of .
If for any two words *a*,*b* of
that join is itself a word
of ,
call it ,
then we say that
is defined at (*a*,*b*).

**Proof:** Suppose not. Then there must exist a projection
with
.
But as before,
must be a projection of ,
giving us a
position (element )
at which the alleged
fails to be
the join of *a* and *b* after all.

Common to these proofs is the existence of a refuting state of ,
which is turned into a refuting state of
by composing it with
*f*. This technique generalizes in the now obvious way to any
relational structure we might impose on .

Now one might deduce from all this that in order to preserve every
possible relational structure we can think of for ,
continuous
functions must be terribly constrained. In some situations this is
indeed the case: for example if
contains a constant word and
does not contain the same constant, there is *no* continuous
function from
to .

But in some situations *every* function from
to
can be
continuous. This happens in particular when
is the dual of the Chu
space
consisting of all words of length *X*. In this case
a projection from
to
is just a word of ,
of length *X*.
But
contains *every* possible word of length *X*. Hence the
test as to whether
is a projection of
must always
succeed, whence every function from
to any Chu space whatsoever
will be continuous. Such an
therefore behaves like a pure set,
an object devoid of structure, which we call a *discrete* Chu space.
This is the Chu counterpart of a discrete topological space.

This way of constructing discrete spaces ensures that no matter what
relational structure we equip
with, and hence ,
all of this
structure may be eliminated by shrinking
enough, namely
to size double-logarithmic (with base )
in the size of
(since
is discrete when *X* is ).

So just how much relational structure *can* we equip
with?
The answer is that it suffices to take all possible relations of all
possible arities on ,
including infinite arities. For arity *X*
(as a set of variables in preference to just a number) these form the
set
of all *X*-ary relations on .
One set *X* of each
cardinality suffices. Most of these relations will be obtainable by
composition from a very small basis. When ,
*X*-ary relations
on 2 for finite *X* are just *X*-ary Boolean operations, for which a
sufficient basis is implication and the constant 0. More generally,
with
still 2, the maximal structure with which we may equip
is that of a complete atomic Boolean algebra. If we ignore questions
of concreteness, this remains true for larger
[Pra95].