T-proximity compatible with T-neighbourhood structure

In this paper, we show that every T-neighbourhood space induces a T-proximity space, where T stands for any continuous triangular norm. An axiom of T-completely regular of T-neighbourhood spaces introduced by Hashem and Morsi (2003) [3], guided by that axiom we supply a Sierpinski object for category T-PS of T-proximity spaces. Also, we define the degree of functional T-separatedness for a pair of crisp fuzzy subsets of a T-neighbourhood space. Moreover, we define the Čech T-proximity space of a T-completely regular T-neighbourhood space, hence, we establishes it is the finest T-proximity space which induces the given T-neighbourhood space.