## Doctoral Dissertations

## Abstract

"This dissertation consists of three subjects: *T*-closed sets, inverse limits with multivalued functions, and hereditarily irreducible maps.

For a subset A of a continuum X define *T*(A) = X \ {x ∈ X : there exists a subcontinuum K of X such that x ∈ int_{x}X(K) ⊂ K ⊂ X \ A}. This function was defined by F. Burton Jones and extensively investigated in the book [20] by Sergio Macias. A subset A of a continuum X is called T-closed set if *T*(A) = A. A characterization of *T*-closed set is given using generalized continua. We also give a counterexample to a hypothesis by David P. Bellamy, Leobardo Fernandez and Sergio Macias about T-closed sets if *T* is idempotent.

We construct a monotone multi-valued bound function *f* : [0; 1]^{2} → 2^{[0,1] 2} such that the inverse limit of the inverse sequence using *f* as the only bounding function is not locally connected. This answers James P. Kelly’s question in [18]. We also give a negative answer to W. T. Ingram’s question in [14]. Precisely, we give an example of an inverse limit sequence on [0,1] with a single upper semi-continuous set-valued bonding function f such that G(*f*^{n}) is an arc for each positive integer n but the inverse limit is not connected.

A mapping *f* : X → Y from a continuum X onto a continuum Y is called hereditarily irreducible if *f*(A) ⊊ *f* (B) for any subcontinua A and B such that A ⊊ B. We investigate properties of hereditarily irreducible maps between continua. Among other things, we introduce a new notion of an order of a point in a continuum that is a bit different than the notion of of an order in the classical sense. The two notions coincide for graphs, but are different in more general locally connected continua. Moreover we prove some theorems about hereditarily irreducible maps with [0,1] as the domain. Thanks to those theorems we may determine if some continua admit hereditarily irreducible maps from [0,1]. We have both necessary conditions and sufficient conditions, so in many cases we may exclude the existence of such maps or prove their existence"--Abstract, page iv.

## Advisor(s)

Charatonik, W. J.

## Committee Member(s)

Roe, Robert Paul

Insall, Matt

Akin, Elvan

Beane, Robbie

## Department(s)

Mathematics and Statistics

## Degree Name

Ph. D. in Mathematics

## Publisher

Missouri University of Science and Technology

## Publication Date

Summer 2017

## Journal article titles appearing in thesis/dissertation

- T-closed sets
- Locally connectedness and inverse limits
- Nonconnected inverse limits
- Hereditarily irreducible maps

## Pagination

viii, 44 pages

## Note about bibliography

Includes bibliographic references.

## Rights

© 2017 Hussam Abobaker, All rights reserved.

## Document Type

Dissertation - Open Access

## File Type

text

## Language

English

## Thesis Number

T 11462

## Electronic OCLC #

1104294854

## Recommended Citation

Abobaker, Hussam, "T-closed sets, multivalued inverse limits, and hereditarily irreducible maps" (2017). *Doctoral Dissertations*. 2754.

https://scholarsmine.mst.edu/doctoral_dissertations/2754