Generalizing the L-fuzzy unit interval

Given an arbitrary connected topological space, an L-fuzzy space is constructed. When the original space is the real line (unit interval) the constructed space is the L-fuzzy real line (L-fuzzy unit interval). For some spaces, including Rⁿ for all n, the original space is embedded as a subspace of the constructed space. Lastly, the construction yields a non-trivial fuzzy topology on certain classically important spaces of monotone mappings.