Document Type
Lecture
Publication Date
1-30-2004
Abstract
Let �� be a topological space, and let ��, �� be families of its subsets. We write �� # �� and say that �� and �� mesh if �� ∩ �� ≠ ∅ for each �� ∈ �� and each �� ∈ ��. �� is compactoid relative to �� if each filter meshing with �� has a cluster point in ��. Let �� be another topological space, and let ��: �� → �� be a set-valued map. �� is said to be upper semicontinuous at �� ∈ �� (usc at ��) if, for each open set �� containing ��(��), there exists a neighborhood �� of �� such that ��(��) ⊆ ��. �� is upper semicontinuous (usc) if it is upper semicontinuous at �� for each �� ∈ ��. The compactoidness of the external filter base seems to be the ultimate strengthening of upper semicontinuity. We will show that it takes place under considerably weaker assumptions about spaces �� and �� than previously thought.
Relational Format
presentation
Recommended Citation
Labuda, Iwo, "On Structure of Upper Semicontinuity" (2004). Analysis Seminar. 58.
https://egrove.olemiss.edu/math_analysis/58
