Closed subspaces ofH-closed spaces

Abstract

The concept of an //-set (a generalization of an //-closed space) was introduced by N. V. Velicko. In this paper we obtain internal properties of //-sets in terms of the Itiadis absolute EX and the Hausdorff absolute PX. Some of the main results are:-An //-closed space JΠs Urysohn iff P~ι(A) is an //-set in PXfor every //-set A c X. 1. Introduction. Throughout this paper all spaces are assumed to be Hausdorff. Our main interests in this paper are the //-closed subspaces and the //-sets of a given space X. Our goal is to characterize these subsets. Our first attempt toward such a characterization is to consider preimages of these subsets in the Iliadis absolute EX and the Hausdorff absolute PX. If X is an //-closed Urysohn space we obtain an answer, namely: A c X is an //-set iff P~\A) c PX is an //-set. For //-closed subspaces of an //-closed Urysohn space the situation is less clear. We give an example of such a space X and an //-closed subspace A such that P~\A) is not //-closed.For the case of non-Urysohn //-closed spaces the situation differs. We present an example of an //-set in a space X which is not the image of a compact subset of EX(oτ an //-set in PX).Although our approach to these questions is quite technical, results 4.7, 4.8, 5.1 and 5.7(i) show that it is rather useful. However, many questions remain unanswered.

A generalization of $ H $-closed spaces

“…All the spaces considered herein are assumed to be Hausdorff. We assume that the reader is familiar with the concepts of H -closedness, H -sets, θ-closed sets and θ-continuity; [12,16] might very well serve as the necessary background. The θ-closure of a subset A of a space X is the set [A] θ ≡ {x ∈ X : U ∩ A = for all open sets U containing x }.…”

“…Some results from literature are cited below: ♯1.1. If f : X → Y is θ-continuous surjective and X is H -closed then Y is H -closed [16].…”

“…Let B be a regular closed subset of a T 2 space X. If A ⊂ X is an H -set and B ⊂ A then B is H -closed [16]. ♯1.10.…”

“…♯1.13. If f : X → Y is θ-continuous and f (X ) ⊂ Z ⊂ Y and Z is dense in Y , then f : X → Z is θ-continuous [16]. ♯1.14.…”

Between compactness and quasicompactness

“…Nearly compact spaces were introduced and studied by Singal and Mathur [24]. Hausdorff almost compact spaces are called H-closed spaces and were introduced by Alexandroff and Urysohn [1] in 1924 and since then have been studied by host of authors (see [4], [8], [15], [16], [19], [20], [21], [22], [30], [31]). The class of quasicompact spaces was initiated by Frolík [5] and has been investigated by Aristotle [3], Stephenson [28], [29] and others.…”