A general theory of fuzzy sets was described employing texture spaces. This work aimed to introduce two concepts in texture theory, namely set-texture and degree-texture spaces using topological spaces and new properties. Moreover, many examples were studied to investigate new texture spaces. The new concepts will use in other topological spaces and find another application on it.

This work is aimed to introduce a new topology on a graph, namely the degree topology.  This topology is defined by the degree of the vertices of the graphs. We find the degree topology for certain types of graphs and determine their types. The degree topology for the complete graph is an indiscrete topology. While The degree topology is generated by a complete bipartite graph  with  is a quasi-discrete topology. In addition, a new property is initiated namely set-  space and discussed the link between it and  space. We verify that every degree topology is a set- space.