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\({\mathcal T}_ 0\)- and \({\mathcal T}_ 1\)-reflections. (English) Zbl 0842.18002
Based on axiomatically given notions of subobject, closure, and point, the author introduces the notions of \(T_1\)- and \(T_0\)-objects in a general categorical setting: \(X\) is a \(T_1\)-object [\(T_0\)-object] if two points \(x\), \(y\) in \(X\) coincide whenever \(x\) belongs to the closure of \(y\) [and \(y\) belongs to the closure of \(x\)]. Under mild assumptions on the category, these objects can be described as the fixed objects of two suitable prereflections [cf. the reviewer, Commun. Algebra 14, 717-740 (1986; Zbl 0587.18002)]. After sufficiently many iterations, these prereflections eventually give the reflectors of the resulting subcategories of \(T_1\)- and of \(T_0\)-objects. In the category \({\mathcal T}op\) of topological spaces, the \(T_0\)-prereflection is already a reflection, while arbitrarily many iterations are needed to describe the \(T_1\)-reflection. However, as the author shows, in other categories the construction of the \(T_0\)-reflector may be as complex as the construction of the \(T_1\)-reflector in \({\mathcal T}op\).
MSC:
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
54A05 Topological spaces and generalizations (closure spaces, etc.)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
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