Heredity and cartesian closed coreflective subcategories of the category of topological spaces

Hereditary Coreflective Subcategories of Categories of Topological Spaces

“…Using Propositions 2, 4 and Theorem 3 we can extend almost all results proved in [11] and [1] Recall (see [11] for details) that if P is a prime space with an accumulation point a, then for each n ∈ N the space P n is defined as follows: P 1 = P and P n+1 = P (P n , a). For each n ∈ N the space P n is a subspace of P n+1 and P ω is the space defined on the set n∈N |P n | such that a subset U is open in P ω if and only if U ∩ P n is open in P n for each n ∈ N. Obviously, for each n ∈ N the space P n is a subspace of P ω and a ∈ P ω .…”

“…If B is a subcategory of Top, then SB denotes the subcategory of Top consisting of all subspaces of spaces that belong to B. It is well known (see, e.g., [8,1]) that if B is a coreflective subcategory of Top, then SB is also coreflective in Top and, clearly, SB = HC(B).…”

“…Moreover, if B is a coreflective subcategory of a quotient-reflective subcategory A of Top and HC(B) is the hereditary coreflective hull of B in Top, then HC(B) ∩ A is the hereditary coreflective hull of B in A. Several results proved in [11] and [1] for (hereditary) coreflective subcategories of Top are extended for (hereditary) coreflective subcategories of an arbitrary (non-trivial) quotient-reflective subcategory of Top. Finally, selected examples of applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional T 2 -spaces, concerning, e.g., the classes of subspaces of corresponding classes of pseudoradial and sequential spaces, are presented.…”

“…In the paper [1] it is proved that a coreflective subcategory of the category Top of topological spaces that contains the class of all finitely-generated spaces is hereditary if and only if it is closed under the formation of prime factors. In this paper we show that an analogous result holds for coreflective subcategories of any epireflective subcategory of Top which is closed under the formation of prime factors.…”

“…In the paper[1] (Corollary 4.6) it is shown that if α and β are regular cardinals and α < β, then the coreflective hull of CH(C(α)) ∪ CH(C(β)) in Top is not hereditary. The corresponding result is obviously valid in any non-trivial quotientreflective subcategory A of Top, i.e.…”