The purposes of this article are to introduce the definitions and typical examples of a k-norm (the inequality is replaced by \(\Vert x+y\Vert \le k(\Vert x\Vert +\Vert y\Vert )\) , \(k\ge 1\) ) and a fuzzy k-norm, which are the generalizations of a norm and a fuzzy norm. And it is shown that there exists a one-to-one correspondence between a fuzzy k-norm and a family of k-norms satisfying some conditions (called a nest of k-norms). What is more, some fuzzifying topological structures induced by a fuzzy k-norm are given, including a fuzzifying neighborhood system, a fuzzifying topology and a fuzzifying topological vector space. Moreover, we conclude that the fuzzifying topology induced by a fuzzy k-norm is exactly the fuzzifying topology induced by its corresponding nest of k-norms, and the fuzzifying topology induced by a nest of k-norms is also exactly the fuzzifying topology induced by its corresponding fuzzy k-norm. Besides, we discuss the relationships among the crisp topologies induced by a fuzzy k-norm and the crisp topologies induced by a nest of k-norms and show them in some summary diagrams.
