Quasihomeomorphisms and Skula spaces

Abstract

Let [Formula: see text] be a topological space. By the Skula topology (or the [Formula: see text]-topology) on [Formula: see text], we mean the topology [Formula: see text] on [Formula: see text] with basis the collection of all [Formula: see text]-locally closed sets of [Formula: see text], the resulting space [Formula: see text] will be denoted by [Formula: see text]. We show that the following results hold: (1) [Formula: see text] is an Alexandroff space if and only if the [Formula: see text]-reflection [Formula: see text] of [Formula: see text] is a [Formula: see text]-space. (2) [Formula: see text] is a Noetherian space if and only if [Formula: see text] is finite. (3) If we denote by [Formula: see text] the Alexandroff extension of [Formula: see text], then [Formula: see text] if and only if [Formula: see text] is a Noetherian quasisober space. We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, [Formula: see text]. A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology [Formula: see text] of [Formula: see text]: The indiscrete components of [Formula: see text] are of the form [Formula: see text], where [Formula: see text] and [Formula: see text] is the intersection of all open sets of [Formula: see text] containing [Formula: see text] (equivalently, [Formula: see text]). We show that [Formula: see text]