REAL FUNCTIONS, COVERS AND BORNOLOGIES
Abstract
The paper surveys the recent results about relationships between
covering properties of a topological space X and the space USC(X) of upper
semicontinuous functions on X with the topology of pointwise convergence.
Dealing with properties of continuous functions C(X) we need shrinkable covers.
The results are extended for A-measurable and upper A-semi-measurable
functions, where A is a family of subsets of X. Similar results for covers respecting
a bornology and spaces USC(X) or C(X) endowed by a topology
dened by using the bornology are presented. Some of them seem to be new.
covering properties of a topological space X and the space USC(X) of upper
semicontinuous functions on X with the topology of pointwise convergence.
Dealing with properties of continuous functions C(X) we need shrinkable covers.
The results are extended for A-measurable and upper A-semi-measurable
functions, where A is a family of subsets of X. Similar results for covers respecting
a bornology and spaces USC(X) or C(X) endowed by a topology
dened by using the bornology are presented. Some of them seem to be new.