Lattices which can be represented as lattices of intervals

We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that every loc-lattice is representable as a lattice of intervals. Furthermore, we provide the complete, unabridged construction for the general representation theorem, establishing that a well-separated lattice is faithfully representable as a lattice of intervals if and only if it is a loc-lattice. Finally, we apply these results to general topology, obtaining novel algebraic characterizations for the bases of weakly orderable and completely orderable topological spaces.