The α-Manifold Space

In this paper we present the definitions of α-Manifold, strongly α- Manifold, s.strongly a-Manifold, s.strongly* α-Manifold, s.strongly** α-Manifold, and we study the relation among them and ordinary manifold, also we give the new definitions of differentiable structure of class C^r about submanifold and we study the relation among them ordinary differential structure of class C^r, and ordinary submanifold.