Upper topology

Upper topology

In mathematics, the upper topology is the least topology defined on a preordered set in which the open sets are the up-sets. The lower topology is defined similarly in terms of the down-sets.

The upper topology on the set {X}=mathbb{R} is defined as {] a,+infty [:ainmathbb{R}cup{pminfty}}. A function on R is upper semi-continuous if and only if it is continuous with respect to the upper topology.

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