In this manuscript, we introduce $LG^{c}$-fuzzy Euclidean topological space in which $L$ denotes a completely distributive lattice with a countable subset dense in it. We use the structure of $LG$-fuzzy topological space $(X,\ \mathfrak{T})$, which $X$ is an $L$-fuzzy subset of the crisp set $M$ and $\mathfrak{T}: L^M_X \to L $, is an $L$-gradation of openness on $X$ to define the fundamental concepts of $LG$-fuzzy analysis such as $LG$-locally compactness and $LG$-paracompactness and prove several theorems. In consequence, we show that any second countable Hausdorff $LG$-fuzzy topological space that is $LG$-locally compact is $LG$-paracompact. Also from any given metric $\rho$ on a crisp set $M$ and $L$-fuzzy subset $X$ of it, we construct an $L$-gradation of openness $\mathfrak{T}_{\rho}$ on $X$ and obtain $LG$-fuzzy topological metric space $(X,\mathfrak{T}_{\rho} )$. Finally, we prove an interesting theorem: Every $LG$-fuzzy topological metric space, is $LG$-paracompact.