We introduce partial dcpo's and show their some applications. A partial dcpo is a poset associated with a designated collection of directed subsets. We prove that (i) the dcpo-completion of every partial dcpo exists; (ii) for certain spaces X, the corresponding partial dcpo's of continuous real valued functions on X are continuous partial dcpos; (iii) if a space X is Hausdorff compact, the lattice of all S-lower semicontinuous functions on X is the dcpo-completion of that of continuous real

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... ed functions on the space; (iv) a topological space has an injective hull iff it is homeomorphic to the pre-Scott space of a continuous partial dcpo whose way-below relation satisfies the interpolation property. 2000 Mathematics Subject Classification. 06B35, 06F30, 06B23, 54C05, 54C30, 54A05.