The notion of Sierpinski space
plays a significant role in general topology. For example, a topological space is 
if and only if it can be embedded into some power of
. Moreover, there exists an analogue of the Sierpinski space for concrete categories under the name of Sierpinski object. Restated in the modern language of category theory, an object 
of a concrete category C is said to be a Sierpinski object provided that for every C-object 
, the hom-set
is an initial source.
Recently, R. Noor, A. K. Srivastava, and S. K. Singh gave a characterization of Sierpinski object in the category of affine bitopological spaces (a bitopological space has two topologies instead of one). Induced by their study, we describe Sierpinski object in the category of composite affine spaces, i.e., spaces, which have a set-indexed family of affine topologies. More precisely, we construct a functor from the category of affine spaces to the category of composite affine spaces, and show a simple condition, under which this functor preserves Sierpinski object. We thus obtain a convenient characterization of composite affine Sierpinski space through Sierpinski affine space, getting, in particular, the affine Sierpinski object of R. Noor et al.
We also suggest a way to obtain similar results for the case of affine systems, namely, to characterize composite affine Sierpinski system through Sierpinski affine system.
This is joint work with Jeffrey T. Denniston (Department of Mathematical Sciences, Kent State University, Kent, Ohio, USA 44242)