The Sorgenfrey line has a locally pathwise connected connectification

Abstract

We answer a question of Alas, Tkacenko, Tkachuk and Wilson by constructing a connected locally pathwise connected Hausdorff space in which the Sorgenfrey line can be densely embedded. A connectification of a T2-space X is a connected Hausdorff space Y in which X can be densely embedded. In such a case Y is called a connectification of X, and X is said to be connectifiable. In 1977 Emeryk and Kulpa showed, in response to a question of van Douwen, that the Sorgenfrey line cannot be densely embedded in a connected T3-space, although it is connectifiable (see [2]). In a recent paper Alas, Tkacenko, Tkachuk and Wilson asked if the Sorgenfrey line has a locally connected connectification ([1]). The aim of this paper is to give a strongly positive answer to the above question. We refer the reader to [3] for notations and terminology not explicitly given. Recall that a collection of pairwise disjoint non-empty open subsets of a space X is called a cellular family. Theorem 1. The Sorgenfrey line has a locally pathwise connected connectification. Proof. Let S = (IR,r) be the Sorgenfrey line and let D = {dn: n E N} be a countable dense subset of S, and for every n, i E N set B(n, i) = [dn, dn + ? [ and C(n, i) = [dn + 1 v dn + B. Let Q be the subset of N3 consisting of all w = (n, m,i) with n 0, then C(n, i + k-1) E YF\; ii) if k < 0, then C(m, i k 1) E TF;; iii) the family 4D = {5X: A E A} is Hausdorff separated (i.e., if A 7 A', then there are F E F2 and F' E YA' such that F f F' = 0). Note that B((n, i) U B(m, i) E F,\. Now let H =]0, 1l[nQ ({n: n E N} U {1n E N}), and set F = Q x H. Received by the editors July 24, 1997 and, in revised form, March 25, 1999. 2000 Mathematics Subject Classification. Primary 54D35, 54D05.