Szymon Dolecki
Active boundaries of upper semicontinuous and compactoid relations; closed and inductively perfect maps
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 54, 51-68 (2000)
MSC:
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
54B20 Hyperspaces
Abstract: It is shown that upper semicontinuity (of a relation) is a weak variant of compactoidness; in
particular, for the relations that are the inverses of maps, the upper semicontinuity amounts to
closedness of the map, and the compactoidness to the perfectness of the map. Therefore the Choquet
theorem on the compactness of active boundaries, and the Vain{\v s}tein lemma on the compactness of
boundaries of fibers are instances of the same quest on conditions that make coincide upper
semicontinuity and compactoidness. The role of Fr\acute{e}chetness and its variants in
this quest is discussed.
Keywords: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.), Hyperspaces