Sergejs Solovjovs – Sierpinski object for composite affine spaces

Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU), Kamycka 129, 16500 Praha 6 - Suchdol, Czech Republic

The notion of Sierpinski space $\mathcal{S}=(\{0,1\},\{\emptyset,\{1\},\{0,1\}\})$ plays a significant role in general topology. For example, a topological space is $T_0$ if and only if it can be embedded into some power of $\mathcal{S}$. 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 $S$ of a concrete category C is said to be a Sierpinski object provided that for every C-object $C$, the hom-set ${\bf C}(C,S)$ 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)