$LG$-FUZZY PARTITION OF UNITY

Abstract

In this paper, we define $ LG^{c}$-fuzzy Euclidean topological space with countable basis, which $L$ denotes a complete distributive lattice and we show that each $LG^{c}$-fuzzy open covering of this space can be refined to an $LG^{c}$-fuzzy open covering that is locally finite. We introduce $C^{\infty }$ $LG$-fuzzy manifold $(X,\ \mathfrak{T}^c)$, with countable basis of $LG$-fuzzy open sets which $X$ is an $L$-fuzzy subset of a crisp set $M$ and $ \ \mathfrak{T} : L^M_X \to L $, is an $L$-gradation of openness on $X$. We prove that for any $LG$-fuzzy topological manifold $(X,\mathfrak{T})$, there exists an $LG$-fuzzy exhaustion. We prove $LG$-Urysohn lemma and also existence of $LG$-partitions of unity on every $LG$-fuzzy topological manifold.