Remainders in Hausdorff compactifications

Abstract

Abstract. We generalize a theorem of Steiner and Steiner and use it to obtain new results concerning remainders for completely regular spaces. Both the locally compact and the nonlocally compact cases are considered.Introduction. The nature of what one added to a completely regular space in order to compactify it has been considered almost from the beginning of the theory. For example, Cech [2] was concerned about the cardinality of /?N\N. It was not until the mid-1960's, however, that the topological properties of remainders was studied in a concerted fashion. Magill, in a series of papers [8], [9], [10] discovered and developed conditions on a space X which would guarantee that members of a certain class of continua would be remainders of X. This theory was extended by Rogers [14] and Chandler [3, 7.8] to conclude that if X is locally compact and not pseudocompact then any compact space containing a dense continuous image of R is a remainder of X. One of the useful results in this area has been a theorem of Steiner and Steiner [15]. Here we generalize this theorem and use it to obtain some new results on remainders. Throughout we follow the general terminology of [3]. In particular, all spaces will be assumed to be completely regular and Hausdorff. A remainder of X is any aX \ X where aX is a compactification

Ordered compactifications with countable remainders

“…A locally compact topological space X has a two-point compactification if and only if X has some compactification with disconnected remainder (for example, 6.16 in [2]). We say an ordered space X is order disconnected if there exists a continuous increasing surjection / : X -» {0,1} where {0,1} has the discrete topology and the usual order 0 < 1.…”

“…An ordered space (X,T,6) is Ti-ordered if 8 is closed in the product space X x X, and is T3.5-ordered (completely regular ordered in [10]) if the following conditions are satisfied: (1) If A C X is closed and x G X\A, then there exist continuous functions /,g : X -• [0,1] with / increasing, g decreasing, f(x) = g(x) = 1, and f(a) A g(a) = 0 for all a € A; [2] (2) If x and y are distinct points in X, then there exists a continuous monotone function / : X -» [0,1] with f(x) -0 and f(y) = 1. Compact T2-ordered implies T3.5-ordered, and T3.s-ordered is hereditary.…”

Compactifications with Discrete Remainders

“…A major problem in compactification theory is to determine when, for each X in a certain class of spaces, there is a member of another class of spaces which can serve as a remainder of X . (See [1], [2], [6], [9], and [13], for example.) The aim of this paper is to characterize the class of spaces X which admit compactifications aX such that aX -X is discrete.…”

Sets of Functions Generating Compactifications via Uniformities. Connectedness

“…[l]- [4], [6], [12] and [13]). Denote by V(X) the family of subsets and y £ ^ Then, for these i,j,k, we have As immediate consequences of 1 and 6 we obtain two versions of Smirnov's theorem: …”