UNIVERZITA KOMENSKÉHO V BRATISLAVE FAKULTA MATEMATIKY, FYZIKY A INFORMATIKY Katedra algebry, geometrie a didaktiky matematiky HEREDITY, HEREDITARY COREFLECTIVE HULLS AND OTHER PROPERTIES OF COREFLECTIVE SUBCATEGORIES OF CATEGORIES OF TOPOLOGICAL SPACES
نویسنده
چکیده
This thesis deals mainly with hereditary coreflective subcategories of the category Top of topological spaces. After preparing the basic tools used in the rest of thesis we start by a question which coreflective subcategories of Top have the property SA = Top (i.e., every topological space can be embedded in a space from A). We characterize such classes by finding generators of the smallest coreflective subcategory of Top with this property. In particular we show that this holds for the category PsRad of the pseudoradial spaces and by modification of the proof of this result we get that the pseudoradial spaces have similar property also in the category Top1 of T1-spaces. This answers a question posed in [AIT]. As the next topic we study hereditary coreflective hull SCH(A) of a prime space A. We succeed to construct a space which generates SCH(A) as a coreflective subcategory of Top. We also provide some results concerning the coreflective subcategories of Top with the property HCK(C) = FG (here HCK(C) denotes the hereditary coreflective kernel of C and FG is the subcategory of finitely generated spaces). We show that the lattice of such subcategories is closed under arbitrary intersection and finite joins, but not under countable joins. In the last two chapters we study a generalization of the original problem. We take an epireflective subcategory A of Top as the base category and we study subcategories of A which are hereditary, additive and divisible in A (i.e., closed under the formation of subspaces, topological sums and quotients with codomain in A; such classes are called more briefly HAD-classes). We show that (under some reasonable conditions on A and B) an additive and divisible class B in A is hereditary if and only if it is closed under the formation of prime factors.
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