A direct approach to K-reflections of T0 spaces

Abstract

In this paper, we provide a direct approach to K-reflections of T 0 spaces. For a full subcategory K of the category of all T 0 spaces and a T 0 space X, let K(X) = {A ⊆ X : A is closed and for any continuousWe call K an adequate category if for any T 0 space X, P H (K(X)) is a K-space. Therefore, if K is adequate, then K is reflective in Top 0 . It is shown that the category of all sober spaces, that of all d-spaces, that of all well-filtered spaces and the Keimel and Lawson's category are all adequate, and hence are all reflective in Top 0 . Some major properties of K-spaces and K-reflections of T 0 spaces are investigated. In particular, it is proved that if K is adequate, then the K-reflection preserves finite products of T 0 spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures.

Some open problems on well-filtered spaces and sober spaces

“…Definition 5.5. ( [49]) Let K be a full category of Top 0 containing Sob and X a T 0 space. A subset A of X is called a K-determined set, provided for any continuous mapping f : X −→ Y to a K-space Y , there exists a unique y A ∈ Y such that f (A) = {y A }.…”

“…The sets in WD(X) are called WD sets. The space X is called a well-filtered determined space, shortly a WD space, if all irreducible closed subsets of X are WD sets, that is, Irr c (X) = WD(X) (see [49,51]).…”

“…Lemma 5.6. ( [49]) Let K be a full category of Top 0 containing Sob and X a T 0 space. Then S c (X) ⊆ KD(X) ⊆ Sob(X) = Irr c (X).…”

“…Question 5.21. ( [49]) Let K be a full subcategory of Top 0 containing Sob (in particular, a Keimel-Lawson category) and X = i∈I X i be the product space of a family {X i : i ∈ I} of T 0 spaces. If each A i ⊆ X i (i ∈ I) is K-determined, must the product set i∈I A i be K-determined?…”

“…Question 5.22. ( [49]) Let K be a full category of Top 0 containing Sob (in particular, a Keimel-Lawson category). Is the product space of an arbitrary family of K-determined spaces K-determined?…”