On hereditary coreflective subcategories of Top

TL;DR

This paper investigates hereditary coreflective subcategories of Top, constructing specific generators and analyzing their properties, especially regarding their kernels and joins, to deepen understanding of their structure.

Contribution

It constructs a prime space generator for the coreflective hull of a non-finitely generated space and analyzes the kernels of joins of such subcategories.

Findings

Constructed a prime space generator with the same cardinality as A.

Showed the hereditary coreflective kernel of joins remains FG.

Provided structural insights into hereditary coreflective subcategories.

Abstract

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.