On some important properties of containwise regularity and category L-CLOSURE

In this work, the multiplication of containwise regular spaces is discussed. Furthermore, the results that the containwise regularity in an induced space is equivalent to the regularity in its underlying space under the condition of closure preserving and the containwise regularity is a good extension in the sense of Lowen are obtained. Moreover, we get an important result that the category L-CLOSURE is a topological category over SET w.r.t. the forgetful function from L-CLOSURE to SET. We obtain another important result that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L -closure spaces which are induced by some crisp closure space.