On hereditary coreflective subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(A \cup B) is again FG

A quasitopos containing CONV and MET as full subcategories

We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a limit function λ which with each filter ℱ associates a map λℱ from the underlying set to the extended positive real line. Continuous maps and contractions can both be characterized as limit function preserving maps

On classes of T0 spaces admitting completions

[EN] For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.Giuli, E. (2003). On classes of T0 spaces admitting completions. Applied General Topology. 4(1):143-155. doi:10.4995/agt.2003.2016.SWORD1431554

Monotone-light factorisation systems and torsion theories

Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that (E, M) induces a monotone-light factorisation system (E',M*) by simultaneously stabilising E and localising M, whenever the torsion theory is hereditary and any object in C is a quotient of an object in X. We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules.Comment: 12 page

On subsequential spaces

AbstractSimple generators for the coreflective category of subsequential spaces, one of them countable, are constructed. Every such must have subsequential order ω1. Subsequentialness is a local property and a countable property, both in a strong sense. A T2-subsequential space may be pseudocompact without being sequential, in contrast to T2-subsequential compact (countably compact, sequentially compact) spaces all being sequential. A compact subsequential space need not be sequential