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|Title||Scott Topology and its Relation to the Alexandroff Topology|
In this thesis, we survey the general topological concepts for the Scott topology, one of the fundamental foundations of theoretical computer science. We shall concentrate on the definition of the T0-Alexandroff space and some of its topological identifications so that the relation between the Scott topology and the T0-Alexandroff topology might be clearly discussed. We introduce here the property of being a T0-space for the Scott topology and due to this we establish the main result for this research that the Scott topology and the T0-Alexandroff topology coincide on finite posets while in general- every Scott open subset is open in T0-Alexandroff topology and the converse need not be true. The Main results of this research: In finite posets, each subset has a top element if and only if it is directed. In finite posets, every ideal is Scott-closed. On any poset X, the Scott topology is a T0-space. A subspace of Scott topology is a Scott subspace. Every algebraic dcpo is continuous. In a continuous finite poset P, no proper subset is a basis for P.
|Publisher||the islamic university|
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