Summary:
One of the most interesting and important problems in topology is the
extension problem: two topological spaces X and Y are given, together with
a closed A X , and we would like to know when a continuous function
g : A ! Y can be extended to a continuous function f from X into Y .
Sometimes there are additional requirements on f, which frequently take
the following form: for every x 2 X, f(x) must be an element of a pre-
assigned subset of Y , which depends on the x. This new problem, which we
call continuous selection problem, is the main issue in this dissertation.
Among the other continuous selection theorems that we refer to, the
most interesting one is the continuous selection theorem for paracompact
space X. The space Y is always Banach. We study this theorem in the 2nd
Chapter where continuous selection theorems for countably paracompact
normal spaces, collectionwise normal spaces and normal spaces are studied
too.
In the 1st Chapter we study some topological properties related to the
2nd Chapter's continuous selection theorems. We study, mostly, paracom-
pact spaces and some of their generalisation such as countably paracompact
spaces and collectionwise normal spaces.
Finally, in the 3rd Chapter we study some applications of the continuous
selection theorems in three insertion theorems and in functional analysis as
well.
Keywords:
Continuous selection, Paracompact space, Insertion theorem, Bartle-Graves Theorem, Kotronis
