Abstract
Tweddle (India J Math 50:99–114, 2008) showed that Banach’s bounded sequence space retains certain properties when endowed with a topology \(\eta \) defined by a maximal almost disjoint (mad) family of natural numbers. In particular, it is Mackey. But it lacks most weak barrelledness properties, forcing Saxon and Tweddle (Adv Math 145:230–238, 1999) to find elsewhere the first example of a Mackey \(\aleph _{0}\)-barrelled space that is not barrelled. We shall show that \(\eta \) varies in tightness as the size of its defining mad family: Tweddle’s spaces are not all isomorphic. But their countable-codimensional subspaces and countable enlargements are all Mackey. Likewise for the Saxon and Tweddle example, which multiplies our scant supply of such. Independently interesting methods are developed.
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