SubArtex Spaces Of an Artex Space Over a Bi-monoid

Abstract

We define SubArtex Space of an Artex space over a  Bi-monoid. We give some examples of SubArtex spaces. We  prove the necessary and sufficient condition for a subset of an Artex space over a bi-monoid to be a SubArtex space. We  prove another equivalent  Proposition for the necessary and sufficient condition for a subset of an Artex space to be a SubArtex space.  We prove a nonempty intersection of two SubArtex spaces of an Artex space over a bi-monoid is  a SubArtex space. Also we prove a nonempty intersection of a family of SubArtex spaces of an Artex space over a bi-monoid is  a SubArtex space.  Finally, we prove, in this chapter, by giving an example, that  the union of two SubArtex spaces need not be a SubArtex space