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In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special properties of their spectra are treated. The concept of thedoi:10.1090/s0002-9947-1978-0515540-7 fatcat:jkofdllhkncqjk2zw3yoysmcfm