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.