Semi-continuous mapping

upper (lower)

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

implies that

where () denotes the limes superior (inferior).

Comments

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)