The internal structure of irreducible continua - Harper, D (2017)
This thesis is an examination of the structure of irreducible continua, with a particular emphasis on local connectedness and monotone maps. A continuum is irreducible if there exist a pair of points such that no proper subcontinuum contains both, with the arc being the most basic example. Being irreducible has a number of interesting implications for a continuum, both locally and globally, and it is these consequences we shall focus on. As mentioned above, the arc is the most straightforward example of an irreducible continuum. Indeed, an intuitive understanding of an irreducible continuum would be that it is structured like an arc, with the points of irreducibility at either end joined by a subspace with no loops or oﬀshoots. In Chapter 2 we will see that for a certain class of continua this intuition is well founded by constructing a monotone map from an irreducible continuum onto an arc. This monotone map will preserve much of the structure of our continuum and as such will provide an insight into that structure. We will next examine a generalisation of irreducibility which considers ﬁnite sets of points rather than just pairs. A number of classical results will be re-examined in this light in Chapter 3. While the majority of these theorems will be shown to have close parallels in higher ﬁnite and inﬁnite irreducibility there will be several which do not hold without further conditions on the continuum. Such anomalies will be particularly prevalent in continua which have indecomposable subcontinua dominating their structure. In Chapter 4 monotone maps will be constructed for ﬁnitely irreducible continua similar to the map to an arc mentioned previously. Chapters 7 and 8 will generalise irreducibility further to the inﬁnite case and we will again construct monotone maps preserving the structure of our continuum. Along with the arc, another highly signiﬁcant irreducible continuum is the sin 1 x continuum. Chapter 5 will focus on this continuum, which will be the basis for a nested sequence of continua. A number of results concerning continuous images of these continua will be presented before using the sequence of continua to deﬁne an indecomposable continuum. This continuum will be investigated, and it will be shown that the union of our nested continua form a composant of the indecomposable continuum. In Chapter 6 we will turn to the question of compactiﬁcations. If a space X is connected then any metric compactiﬁcation of X will be a continuum. This chapter will answer the question of when a compactiﬁcaion is an irreducible continuum, with the remainder of the compactiﬁcation consisting of all of the irreducible points. A list of properties will given such that a continuum has such a compactiﬁcation if and only if it has each property on the list. It will also be demonstrated that each of these properties is independent of the others. Finally, in Chapter 9 we will revisit the idea of structure-preserving monotone maps, but this time in continua which are not irreducible. Motivated by the ﬁbres of the maps in previous chapters, we will introduce two categories of subcontinua of a continuum X. The ﬁrst will be nowhere dense subcontinua which are maximal with this property and the second will be subcontinua about which X is locally connected and which are minimal with this property. Continua in which every point lies in a maximal nowhere dense subcontinuum will be examined, as well as spaces in which every point lies in a unique minimal subcontinuum about which X is locally connected. We will also look at the properties of monotone maps arising from partitions of X into such subcontinua, and will prove that if every point of X lies in a maximal nowhere dense subcontinuum then the resulting quotient space will be one dimensional.