# Semi-continuous mapping

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upper (lower)

A mapping from a topological space into a partially ordered set such that

implies that

where () denotes the limes superior (inferior).

On a partially ordered set the collection consisting of and all sets is a base for a topology on , denoted by , and and all sets define a topology . The mapping is upper semi-continuous, (u.s.c.) (respectively, lower semi-continuous (l.s.c.)) if and only if (respectively, ) is continuous.

In fact, upper and lower semi-continuity are usually defined only for mappings to the real line . In terms of open sets, one sees that is upper (lower) semi-continuous if and only if () is open for every .

Semi-continuity is also defined for set-valued mappings. A mapping is upper (lower) semi-continuous if for every open subset of the set (the set ) is open.

Note that if a mapping is regarded as a set-valued mapping , , then is lower semi-continuous if and only if is lower semi-continuous; and is upper semi-continuous if and only if is upper semi-continuous.

#### References

 [a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Semi-continuous mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_mapping&oldid=13587
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article