Postgraduate Thesis uoadl:1317970 554 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2012-09-07

2012

Κοτρώνης Στυλιανός

Γεώργιος Κουμουλλής Καθηγ. (Επιβλέπων), Σοφοκλής Μερκουράκης Καθηγ., Αθανάσιος Τσαρπαλιάς Καθηγ.

Θεωρήματα συνεχούς επιλογής και παρασυμπαγείς τοπολογικοί χώροι

Greek

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.

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.

Continuous selection, Paracompact space, Insertion theorem, Bartle-Graves Theorem, Kotronis

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