# INTUITIONISTIC TOPOLOGICAL SPACES WITH $L$-GRADATIONS OF OPENNESS AND NONOPENNESS WITH RESPECT TO $LT$-NORM $T$ AND $LC$-CONORM $C$ ON $X$

## Keywords:

Spiral $LT$-norm, intuitionistic $L$-fuzzy subset, intuitionistic $L$-fuzzy subgroup with respect to the norm $L(T,C)$-norm, intuitionistic $L$-gradation of closeness and noncloseness with respect to $L(T,C)$-norm.## Abstract

In this paper, we assume that $L= <L,\leq,\bigwedge,\bigvee, '>$ is a complete distributive lattice set with at least 2 elements and $(L,+)$ is also an additive group. We introduce an $LT$-norm $T$ and an $LC$-conorm $C$ on the lattice set $L$ (briefly $L(T,C)$-norm). Furthermore using this norm, we define spiral $LT$-norm and spiral $LC$-conorm of any countable sequence in $L$. Also we introduce $IL(T,C)$-gradations of openness on $X$ which $X$ is an $L$-fuzzy subset of a nonempty set $M$ and we prove that the set of all $IL(T,C)$-gradations of openness on $X$ is a semicomplete lattice. We introduce intuitionistic $L$-fuzzy topological space with $L$-gradation of openness and nonopenness with respect to the $L(T,C)$-norm ( briefly $ILG(T,C)$-fuzzy topological space). As an example we define an $IL(T,C)$-fuzzy subspace of $\Lambda \mathbb{R}^{m}$, the exterior algebra on $\mathbb{R}^{m}$.