Semi-continuous mapping
upper (lower)
A mapping from a topological space
into a partially ordered set
such that
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implies that
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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) |
Semi-continuous mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_mapping&oldid=48656