Two constructions of continua: inverse limits and compactificationsTina Sovič
, 2013, doktorska disertacija
Opis: In the thesis we talk about two different constructions of continua. First we present the generalized inverse limits, with help of which we construct Wazewski's universal dendrite. What follows is a description of the compactifications of a ray and the presentation of results about their span.
The first chapter will be an introduction to the continuum theory trough interesting examples, as sin(1/x)-continuum, Hilbert cube, Brouwer-Janiszewski-Knaster continuum and pseudoarc. We will present some of their properties, among which irreducibility, smoothness and span zero are the most important ones for us.
In the continuation we intend to present some various constructions of continua. The main focus will be on the generalized inverse limits and compactifications of rays, which will also be a central part of the thesis. In this chapter, we also study inverse limits in the category of compact Hausdorff spaces with upper semi-continuous functions. We show that the inverse limits with upper semi-continuous bonding functions, together with the projections are weak inverse limits in this category.
The following two are the most important chapters in the thesis. The first is a detailed description of a construction of the family of upper semi-continuous functions f, such that the inverse limit of the inverse sequence of unit intervals and f, as the only bonding function, is homeomorphic to Wazewski's universal dendrite for each of it. Among other results we will also give a complete characterization of comb-functions, for which the inverse limits of the type described above are dendrites.
The next important chapter will be about compactifications of rays. In the first part of this chapter we will use compactifications to prove that for each continuum Y there is an irreducible smooth continuum that contains a topological copy of Y. The second part presents the main results of this chapter; i.e. the span of a compactification of a ray with a remainder that has a span zero is also zero. In the proofs of this chapter we will help ourselves with a discretization of span.
Ključne besede: continua, inverse limit, inverse sequence, upper semi-continuous function, set-valued functions, bonding function, hyperspace, dendrite, universal dendrite, category, compactification, compactification of a ray, smooth continua, irreducible continua, span, span zero
Objavljeno: 25.09.2013; Ogledov: 1313; Prenosov: 84
Celotno besedilo (861,84 KB)