A hereditary class on a set X is a nonempty collection of subsets of X which is closed under subsets. In this paper, we present a new structure of proximity spaces by using a hereditary class, called H-proximity spaces, as a generalization of Efremoviˇ, c proximity spaces, I-proximity spaces and coarse proximity spaces. Some properties of this proximity structure and generalized topology induced by it are studied.

It is well-known that M. Matsumoto and R. Miron constructed an orthonormal frame for an n-dimensional Finsler space and the frame was called ’Miron frame’. Then, T. N. Pandey and D. K. Diwedi and P. N. Pandey and Manish Gupta studied four dimensional Finsler spaces in terms of scalars. P. N. Pandey and Manish Gupta also studied four dimensional Berwald space with vanishing h-connection vector. Gauree Shankar, G C. Chaubey and Vinay Pandey studied the main scalar of a five-dimensional Finsler space. In the present paper, we study a five dimensional Berwald space with vanishing hconnection vector hi, Ji, ki.

In this paper, we consider the problem of isometric extension in the unit sphere of the space sp(a, H). We obtain that Lamperti isometric mapping of the unit sphere S(sp(a, H)) into itself can be extended to an isometry on the whole space sp(a, H).