Compactification-like extensions

Abstract

Abstract. Let X be a space. A space Y is called an extension of X if Y contains X as a dense subspace. For an extension Y of X the subspace Y \X of Y is called the remainder of Y . Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X pointwise. For two (equivalence classes of) extensions Y and Y ′ of X let Y ≤ Y ′ if there is a continuous mapping of Y ′ into Y which fixes X pointwise. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P. If P is compactness then P-extensions are called compactifications.The aim of this article is to introduce and study classes of extensions (which we call compactification-like P-extensions, where P is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like P-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We will then consider the classes of compactification-like P-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like P-extensions of a space among all its Tychonoff P-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like Pextensions of a Tychonoff space X and the topology of a certain subspace of its outgrowth βX\X. We conclude with some applications, including an answer to an old question of S. Mrówka and J.H. Tsai: For what pairs of topological properties P and Q is it true that every locally-P space with Q has a one-point extension with both P and Q? An open question is raised.

Topological extensions with compact remainder

“…The following lemma, which is the counterpart of Lemmas 2.5 and 3.4, is a slight modification of Lemma 2.10 of [10]. Lemma 4.8.…”

“…The results of this part are from [13] (for a proof of Lemma 2.2, see [10]); we include them here for completeness of results and reader's convenience. Definition 2.1.…”

“…, the Lindelöf property (more generally, the µ-Lindelöf property), paracompactness, metacompactness, countable paracompactness, subparacompactness, submetacompactness (or θ-refinability), the σ-para-Lindelöf property and also α-boundedness. (See [10] for the proofs and [2], [24] and [25] for the definitions. )…”

“…Analogously, we show that the converse of Theorem 4.12 (the dual result of Theorems 2.20 and 3.9) does not hold either. This will be done through an example (Example 4.26) for a specific choice of a compactness-like topological property P. The example (which shares several ideas of the proof of Theorem 4.36 of [10] and certain results from [11]; e.g. Lemmas 2.10, 4.1 and 4.3 of [11]) is very technical, and requires us to state and prove a series of lemmas preceding it.…”

“…The following subspace of βX, introduced and studied in [10] (see also [11], [13] and [14]), plays a crucial role in what follows. Definition 4.5.…”