Let us now consider
with
and
We may note in passing that neither of the Boolean recombination spaces
are topological spaces, even after adding the empty set to
.
Note that a pair
with
is a topological space if three conditions are met:
2. 
and
3. 
Neither 
nor
contain T and are thus disqualified as topological spaces.
because 
if 
,
but the third condition is not met by any of the recombination spaces.
For example take A and B to be non-overlapping recombination sets,
then 
can not be a recombination set, i.e. there are no
for which 
this is the recombination set.
Now we want to construct the recombination space for Boolean vectors, which will lead to the following result:
Proposition: The hypercube is embedded in all three recombination spaces
In other words, the point mutation space for bit strings is homomorphic with the corresponding recombination spaces.
Let 
be the number of unordered pairs of elements in C(n) whose Hamming distance is x. Then
takes on values as follows:
To begin our construction we identify each of the
singleton sets
with a vertex of the hypergraph. Consider now the pairs of vertices
having Hamming distance one. Then for
,
we have
.
Thus in each recombination hypergraph an edge of rank 2 exists between each
pair of vertices separated by a Hamming distance of one. There are
such pairs in C(n). These are the only rank-2 edges in each graph.
This implies that in each case, the recombination space contains an
isomorphic copy of the well-known n-dimensional hypercube of point mutation
space.
Let us conditionally accept the hypercube as the support
structure of each space
.
We must still investigate the remaining edges. Given two vertices a and b
with Hamming distance 2, then in general
Note that
Therefore 
is a two dimensional face (square) on the n-dimensional hypercube.
In fact, the same face will be generated exactly twice, since
.
Thus there are 
generalized edges of order 4, each corresponding to a two-dimensional face
of the n-dimensional hypercube.
We may go no further in a simultaneous investigation of the three
recombination spaces
,
since strings at Hamming distance greater than 2 generate different
recombination sets under the three operators. It has been noted elsewhere
that under free recombination every pair of elements generates a hypercube
of dimension equal to the Hamming distance between them [8]. With this in
mind, and recalling that
(see section 3) we will investigate the minimal recombination space
generated by one-point crossover.
We many note that a square is a circumference of a two-dimensional hypercube. We generalize this fact for all edges of the space
in the following Lemma:
Lemma:
is a circumference of the
- dimensional hypercube
, which is embedded in the n-dimensional
hypercube.
With this result one may construct the 
recombination sets of order 2x, for x=2, ... ,n.
Figure 1 depicts a generalized recombination set
with d(a,b)=k.
Note that vertices connected in the recombination set are in fact also
connected in the corresponding hypercube of the point mutation space. The
hypercube thereby provides a natural support structure of the space
Incidentally the recombination
spaces have the same metric as the point-mutation spaces.
At the end of this section we want to add a few notes on strings (or vectors) with more than two possible states on each position. It will be shown that most of what has been shown in the Boolean case is also true for the multi-allele case.
Let 
be arbitrary sets of cardinality >1. Then we may define a multi-allele
type space as the vector space over 
So
The definitions of the recombination operators remain the same as in the
Boolean case, and the recombination spaces
are defined analogously.
Recombination operators are binary. A (local) consideration of
is essentially an examination of only up to two alleles at the n positions.
With this in mind, it is natural to define a distance on
equal to the Hamming distance. The expressions for the orders of
recombination sets that are identical to the ones obtained in the Boolean
case. Again the sense of neighborhood is the same as in the corresponding
point mutation space.
Figure 1: Topological relationships among the elements of the one-point crossover
recombination set
. The elements connected by an
edge all have a pair wise Hamming distance of one. The graph represents the circumference of the hypercube with the diameter H=d(a,b).