For the purpose of our constructions, we will assume the existence of a
reasonably large set 
so that
each family
is locally finite and 
is a 
-locally
finite base for a topology on F(X,L).
The metrics 
of F(X,L) can now be defined as
constructions in [5] related to proving the metrization theorem:
where
and
It can be shown that
is a (pseudo-)metric space for all 
.
Since 
, it is intuitively clear that
the 'gap' between U' and the complement of U
closes up as r approaches s. We will use 
,
to denote the intuitive counterpart of ,
where the 'gap' between U' and the complement of U is assumed
to vanish or to be neglible.
Thus, we arrive at our proposal for a metric on F(X,L):
where 
is the cardinality for a set.
For computing purposes we will assign 
to the following:
where 
, 
. This is a fair
approximation, provided we consider 
and
as 2-dimensional discs with radii 
. The activation
is the non-overlapping area of the two circles normalized to the
[0,1] interval. If and
are identical, there is no non-overlapping area and the distance is
zero.
If the distance between 
and 
is more than 
there will be no overlapping
area and reaches its maximum value.
The metric is based on an activation of 
which implies that if is a metric then is too. This
property is guaranteed by the activation function.
We now have that is a pseudo-metric on F(X,L),
and it is obvious that is not a metric, since this
would imply that 
iff 
and the whole
construction with 
would be of no use.
The granularity of the pseudo-metric is provided by the
size of the neighbourhoods 
within F(Y,M), as desired.