Local connectedness and other properties of GA compactifications

Abstract

Indagationes Mathematicae (Proceedings) 75 (1972) 11-18. doi:10.1016/1385-7258(72)90022-4Received by publisher: 0000-01-01Harvest Date: 2016-01-04 12:23:34DOI: 10.1016/1385-7258(72)90022-4Page Range: 11-1

Dendritic compactification

“…From Lemma 2 of [11], the collection £f is closed under finite intersections. The first two assertions now follow from Theorem 6 of [5], and third assertion follows from Theorem 4 of [5]. Proof.…”

“…is identical to SK By Lemma 4.2, ^ is a regular and normal closed subbase for X, and by Lemma 4.3, ^~ = {C~: Cê } is a closed subbase for X* which satisfies the condition of subbase regularity. It follows now by Theorem 7 of [5] that X* is locally connected.…”

“…Such compactifications are called GA compactifications, and their general properties have been studied by de Groot, Hursch, and Jensen in [5], [6], and [7]. In §4 below, it is shown that the dendritic compactification of a rim compact dendritic space is a GA compactification.…”

Every Hausdorff Compactification of a Locally Compact Separable Space is a Ga Compactification

“…In [4], De Groot and Aarts constructed Hausdorff compactifications of topological spaces to obtain a new intrinsic characterization of complete regularity. These compactifications were called GA compactifications in [5] and [7]. A characterization of complete regularity was earlier given by Frink [3], by means of Wallman compactifications, a method which led to the intriguing problem of whether every Hausdorff compactification is a Wallman compactification.…”