Infinite cartesian products of differentials and frolicher lie groups

Given a set, the Sikorski differential space structure is determined on it by a collection of
real-valued functions, while the Frölicher (smooth) structure is defined by a pair of paths
into along with real-valued functions which fulfil specified sets of axioms. According to A.
Batubege and P. Ntumba, when these structures are provided with an additional group
operation that is compatible with the smooth structure, they are then called differential
groups or Frölicher Lie groups, respectively. The infinite Cartesian product of differential
groups was investigated by W. Sasin, and since a Frölicher space is a differential space in
the sense of Sikorski, it turns out that a Frölicher Lie group is a differential group. Now, the
differential structure on the product of differential groups is the product of structures of the
factors. On the product of Frölicher Lie groups as for general Frölicher spaces, it is rather the
set of structure curves that has this property, and not the set of structure functions. In this
study we use a class of Frölicher spaces free of this defect, in order for the resulting Frölicher
Lie groups to satisfy the property similar to that of smooth functions on the product of
differential groups. To this end, we consider a class of differential groups made of differential
spaces whose set of structure functions is reflexive in the sense that it generates Frölicher
curves from which the generated Frölicher functions are exactly the Sikorski functions which
induced the smooth structure. Such differential spaces, so-called pre-Frölicher spaces by
A. Batubege, induce a class of Frölicher spaces (Frölicher Lie groups) so-called DF-spaces
(groups) on which, unlike differential groups, the set of smooth functions is the product of
sets of structure functions from the factors. This induces the results similar to the study by
W. Sasin.